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S.Duplij, Arity shape of polyadic algebraic structures (version 2)

Steven Duplij (Stepan Douplii), profile picture
Steven Duplij (Stepan Douplii)

This document explores polyadic algebraic structures, particularly emphasizing the arity shape and restrictions of operations within those structures. It introduces key concepts such as polyadic vector spaces, dual spaces, and applications in number theory and operator theory, including discussions on polyadic analogs of well-known conjectures. The paper also provides applications to polynomial equations, establishing new results and insights into the nature of polyadic numbers and their mathematical implications.

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arXiv:1703.10132v2[math.RA]1May2017 ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES STEVEN DUPLIJ ABSTRACT. Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and lead to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, oper- ator norms, isometries and projections, as well as polyadic C∗ -algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with num- ber theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem are formulated. For the nonderived polyadic ring operations (polyadic numbers) neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov’s theorem for the Tarry-Escott problem is presented. CONTENTS INTRODUCTION 2 1. ONE SET POLYADIC “LINEAR” STRUCTURES 3 1.1. POLYADIC DISTRIBUTIVITY 3 1.2. POLYADIC RINGS AND FIELDS 4 2. TWO SET POLYADIC STRUCTURES 5 2.1. POLYADIC VECTOR SPACES 5 2.2. ONE-SET POLYADIC VECTOR SPACE 8 2.3. POLYADIC ALGEBRAS 9 3. MAPPINGS BETWEEN POLYADIC ALGEBRAIC STRUCTURES 12 3.1. POLYADIC FUNCTIONALS AND DUAL POLYADIC VECTOR SPACES 13 3.2. POLYADIC DIRECT SUM AND TENSOR PRODUCT 15 4. POLYADIC INNER PAIRING SPACES AND NORMS 19 APPLICATIONS 21 5. ELEMENTS OF POLYADIC OPERATOR THEORY 21 5.1. MULTISTARS AND POLYADIC ADJOINTS 23 5.2. POLYADIC ISOMETRY AND PROJECTION 27 5.3. TOWARDS POLYADIC ANALOG OF C∗-ALGEBRAS 28 6. CONGRUENCE CLASSES AS POLYADIC RINGS 30 6.1. POLYADIC RING ON INTEGERS 31 6.2. LIMITING CASES 33 7. EQUAL SUMS OF LIKE POWERS DIOPHANTINE EQUATION OVER POLYADIC INTEGERS 35 7.1. POLYADIC ANALOG OF THE LANDER-PARKIN-SELFRIDGE CONJECTURE 36 7.2. FROLOV’S THEOREM AND THE TARRY-ESCOTT PROBLEM 40 ACKNOWLEDGMENTS 41 REFERENCES 42 LIST OF TABLES 43 Date: March 29, 2017. 2010 Mathematics Subject Classification. 11D41, 11R04, 11R06, 17A42, 20N15, 47A05, 47L30, 47L70, 47L80.
2 STEVEN DUPLIJ INTRODUCTION The study of polyadic (higher arity) algebraic structures has a two-century long history, start- ing with works by Cayley, Sylvester, Kasner, Pr¨ufer, D¨ornte, Lehmer, Post, etc. They took a sin- gle set, closed under one (main) binary operation having special properties (the so called group- like structure), and “generalized” it by increasing the arity of that operation, which can then be called a polyadic operation and the corresponding algebraic structure polyadic as well1 . An “ab- stract way” to study polyadic algebraic structures is via the use of universal algebras defined as sets with different axioms (equational laws) for polyadic operations COHN [1965], GR ¨ATSER [1968], BERGMAN [2012]. However, in this language some important algebraic structures cannot be de- scribed, e.g. ordered groups, fields, etc. DENECKE AND WISMATH [2009]. Therefore, another “concrete approach” is to study examples of binary algebraic structures and then to “polyadize” them properly. This initiated the development of a corresponding theory of n-ary quasigroups BELOUSOV [1972], n-ary semigroups MONK AND SIOSON [1966], ZUPNIK [1967] and n-ary groups GAL’MAK [2003], RUSAKOV [1998] (for a more recent review, see, e.g., DUPLIJ [2012] and comprehensive list of references therein). The binary algebraic structures with two operations (addition and multiplication) on one set (the so-called ring-like structures) were later on generalized to (m, n)-rings CELAKOSKI [1977], CROMBEZ [1972], LEESON AND BUTSON [1980] and (m, n)- fields IANCU AND POP [1997], while these were investigated mostly in a more restrictive manner by considering particular cases: ternary rings (or (2, 3)-rings) LISTER [1971], (m, 2)-rings BOCCIONI [1965], POP AND POP [2002], as well as (3, 2)-fields DUPLIJ AND WERNER [2015]. In the case of one set, speaking informally, the “polyadization” of two operations’ “interaction” is straightforward, giving only polyadic distributivity which does not connect or restrict their arities. However, when the number of sets becomes greater than one, the “polyadization” turns out to be non- trivial, leading to special relations between the operation arities, and introduces additional (to arities) parameters, which allows us to classify them. We call the selection of such relations an arity shape and formulate the arity partial freedom principle that not all arities of operations during “polyadization” of binary operations are possible. In this paper we consider two-set algebraic structures in the “concrete way” and provide the con- sequent “polyadization” of binary operations on them for the so-called module-like structures (vector spaces) and algebra-like structures (algebras and inner product spaces). The “polyadization” of the binary scalar multiplication is made in terms of multiactions introduced in DUPLIJ [2012], having special arity shapes parametrized by the number of intact elements (ℓid) in the corresponding multi- actions. Then we “polyadize” the related constructions, as dual vector spaces and direct sums, and also tensor products, and show that, as opposed to the binary case, they can be implemented in spaces of different arity signatures. The “polyadization” of inner product spaces and related norms gives additional arity shapes and restrictions. In the resulting TABLE 2 we present the arity signatures and shapes of the polyadic algebraic structures under consideration. In the application part we note some starting points for polyadic operator theory by introducing multistars and polyadic analogs of adjoints, operator norms, isometries and projections. It is proved (Theorem 5.7) that, if the polyadic inner pairing (the analog of the inner product) is symmetric, then all multistars coincide and all polyadic operators are self-adjoint (as opposed to the binary case). The polyadic analogs of C∗ -algebras, Toeplitz algebras and Cuntz algebras are presented in terms of the polyadic operators introduced here, and the ternary example is given. Another application is connected with number theory: we show that the internal structure of the congruence classes is described by a polyadic ring having a special arity signature (TABLE 3), and we call them polyadic integers (numbers) Z(m,n) (Definition 6.17). They are classified by polyadic shape invariants, and the relations between them giving the same arity signature are established. Also the 1 We use the term “polyadic” in this sense only, while there are other uses in the literature (see, e.g., HALMOS [1962]).
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 3 limiting cases are analyzed, and it is shown that in one such a case the polyadic rings can be embedded into polyadic fields with binary multiplication, which leads to the so-called polyadic rational num- bers CROMBEZ AND TIMM [1972]. Then we consider the Diophantine equations over these polyadic rings in a straightforward manner: we change only arities of operations (“additions” and “multipli- cations”), but save their mutual “interaction”. In this way we try to “polyadize” the equal sums of like powers equation and formulate the polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem LANDER ET AL. [1967]. It is shown, that in the simplest case, when the polyadic “addition” and “multiplication” are nonderived (e.g., polyadic numbers), neither conjecture is valid, and counterexamples are presented. Finally, we apply Frolov’s theorem for the Tarry-Escott problem DORWART AND BROWN [1937], NGUYEN [2016] over polyadic rings to obtain new solu- tions to the equal sums of like powers equation for a fixed congruence classes. 1. ONE SET POLYADIC “LINEAR” STRUCTURES We use concise notations from our previous work on polyadic structures DUPLIJ [2012, 2016]. Take a non-empty set A, then n-tuple (or polyad) consisting of the elements (a1, . . . , an), ai ∈ A, is denoted by bold letter (a) taking it values in the Cartesian product A×n . If the number of elements in the n-tuple is important, we denote it manifestly a(n) , and an n-tuple with equal elements is denoted by (an ). On the Cartesian product A×n one can define a polyadic operation µn : A×n → A, and use the notation µn [a]. A polyadic structure A is a set A which is closed under polyadic operations, and a polyadic signature is the selection of their arities. For formal definitions, see, e.g., COHN [1965]. 1.1. Polyadic distributivity. Let us consider a polyadic structure with two operations on the same set A: the “chief” (multiplication) n-ary operation µn : An → A and the additional m-ary operation νm : Am → A, that is A | µn, νm . If there are no relations between µn and νm, then nothing new, as compared with the polyadic structures having a single operation A | µn or A | νm , can be said. Informally, the “interaction” between operations can be described using the important relation of distributivity (an analog of a · (b + c) = a · b + a · c, a, b, c ∈ A in the binary case). Definition 1.1. The polyadic distributivity for the operations µn and νm (no additional properties are implied for now) consists of n relations µn [νm [a1, . . . am] , b2, b3, . . . bn] = νm [µn [a1, b2, b3, . . . bn] , µn [a2, b2, b3, . . . bn] , . . . µn [am, b2, b3, . . . bn]] (1.1) µn [b1, νm [a1, . . . am] , b3, . . . bn] = νm [µn [b1, a1, b3, . . . bn] , µn [b1, a2, b3, . . . bn] , . . . µn [b1, am, b3, . . . bn]] (1.2) ... µn [b1, b2, . . . bn−1, νm [a1, . . . am]] = νm [µn [b1, b2, . . . bn−1, a1] , µn [b1, b2, . . . bn−1, a2] , . . . µn [b1, b2, . . . bn−1, am]] , (1.3) where ai, bj ∈ A. It is seen that the operations µn and νm enter into (1.1)-(1.3) in a non-symmetric way, which allows us to distinguish them: one of them (µn, the n-ary multiplication) “distributes” over the other one νm, and therefore νm is called the addition. If only some of the relations (1.1)-(1.3) hold, then such distributivity is partial (the analog of left and right distributivity in the binary case). Obviously, the operations µn and νm need have nothing to do with ordinary multiplication (in the binary case denoted by µ2 =⇒ (·)) and addition (in the binary case denoted by ν2 =⇒ (+)), as in the below example.
4 STEVEN DUPLIJ Example 1.2. Let A = R, n = 2, m = 3, and µ2 [b1, b2] = bb2 1 , ν3 [a1, a2, a3] = a1a2a3 (product in R). The partial distributivity now is (a1a2a3)b2 = ab2 1 ab2 2 ab2 3 (only the first relation (1.1) holds). 1.2. Polyadic rings and fields. Here we briefly remind the reader of one-set (ring-like) polyadic structures (informally). Let both operations µn and νm be (totally) associative, which (in our defi- nition DUPLIJ [2012]) means independence of the composition of two operations under placement of the internal operations (there are n and m such placements and therefore (n + m) corresponding relations) µn [a, µn [b] , c] = invariant, (1.4) νm [d, νm [e] , f] = invariant, (1.5) where the polyads a, b, c, d, e, f have corresponding length, and then both A | µn | assoc and A | νm | assoc are polyadic semigroups Sn and Sm. A commutative semigroup A | νm | assoc, comm is defined by νm [a] = νm [σ ◦ a], for all σ ∈ Sn, where Sn is the symmetry group. If the equation νm [a, x, b] = c is solvable for any place of x, then A | νm | assoc, solv is a polyadic group Gm, and such x = ˜c is called a (additive) querelement for c, which defines the (additive) unary queroperation ˜ν1 by ˜ν1 [c] = ˜c. Definition 1.3. A polyadic (m, n)-ring Rm,n is a set A with two operations µn : An → A and νm : Am → A, such that: 1) they are distributive (1.1)-(1.3); 2) A | µn | assoc is a polyadic semigroup; 3) A | νm | assoc, comm, solv is a commutative polyadic group. It is obvious that a (2, 2)-ring R2,2 is an ordinary (binary) ring. Polyadic rings have much richer structure and can have unusual properties CELAKOSKI [1977], CROMBEZ [1972], ˇCUPONA [1965], LEESON AND BUTSON [1980]. If the multiplicative semigroup A | µn | assoc is commutative, µn [a] = µn [σ ◦ a], for all σ ∈ Sn, then Rm,n is called a commutative polyadic ring, and if it contains the identity, then Rm,n is a (polyadic) (m, n)-semiring. If the distributivity is only partial, then Rm,n is called a polyadic near-ring. Introduce in Rm,n additive and multiplicative idempotent elements by νm [am ] = a and µn [bn ] = b, respectively. A zero z of Rm,n is defined by µn [z, a] = z for any a ∈ An−1 , where z can be on any place. Evidently, a zero (if exists) is a multiplicative idempotent and is unique, and, if a polyadic ring has an additive idempotent, it is a zero LEESON AND BUTSON [1980]. Due to the distributivity (1.1)-(1.3), there can be at most one zero in a polyadic ring. If a zero z exists, denote A∗ = A {z}, and observe that (in distinction to binary rings) A∗ | µn | assoc is not a polyadic group, in general. In the case where A∗ | µn | assoc is a commutative n-ary group, such a polyadic ring is called a (polyadic) (m, n)-field and Km,n (“polyadic scalars”) (see LEESON AND BUTSON [1980], IANCU AND POP [1997]). A multiplicative identity e in Rm,n is a distinguished element e such that µn a, en−1 = a, (1.6) for any a ∈ A and where a can be on any place. In binary rings the identity is the only neutral element, while in polyadic rings there can exist many neutral (n − 1)-polyads e satisfying µn [a, e] = a, (1.7) for any a ∈ A which can also be on any place. The neutral polyads e are not determined uniquely. Obviously, the polyad (en−1 ) is neutral. There exist exotic polyadic rings which have no zero, no identity, and no additive idempotents at all (see, e.g., CROMBEZ [1972]), but, if m = 2, then a zero always exists LEESON AND BUTSON [1980].
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 5 Example 1.4. Let us consider a polyadic ring R3,4 generated by 2 elements a, b and the relations µ4 a4 = a, µ4 a3 , b = b, µ4 a2 , b2 = a, µ4 a, b3 = b, µ4 b4 = a, (1.8) ν3 a3 = b, ν3 a2 , b = a, ν3 a, b2 = b, ν3 b3 = a, (1.9) which has a multiplicative idempotent a only, but has no zero and no identity. Proposition 1.5. In the case of polyadic structures with two operations on one set there are no con- ditions between arities of operations which could follow from distributivity (1.1)-(1.3) or the other relations above, and therefore they have no arity shape. Such conditions will appear below, when we consider more complicated universal algebraic struc- tures with two or more sets with operations and relations. 2. TWO SET POLYADIC STRUCTURES 2.1. Polyadic vector spaces. Let us consider a polyadic field KmK ,nK = K | σmK , κnK (“polyadic scalars”), having mK-ary addition σmK : KmK → K and nK-ary multiplication κnK : KnK → K, and the identity eK ∈ K, a neutral element with respect to multiplication κnK enK −1 K , λ = λ, for all λ ∈ K. In polyadic structures, one can introduce a neutral (nK − 1)-polyad (identity polyad for “scalars”) eK ∈ KnK−1 by κnK [eK, λ] = λ. (2.1) where λ ∈ K can be on any place. Next, take a mV -ary commutative (abelian) group V | νmV , which can be treated as “polyadic vectors” with mV -ary addition νmV : VmV → V. Define in V | νmV an additive neutral element (zero) zV ∈ V by νmV zmV −1 V , v = v (2.2) for any v ∈ V, and a “negative vector” ¯v ∈ V as its querelement νmV [aV , ¯v, bV ] = v, (2.3) where ¯v can be on any place in the l.h.s., and aV , bV are polyads in V. Here, instead of one neutral element we can also introduce the (mV − 1)-polyad zV (which may not be unique), and so, for a zero polyad (for “vectors”) we have νmV [zV , v] = v, ∀v ∈ V, (2.4) where v ∈ V can be on any place. The “interaction” between “polyadic scalars” and “polyadic vectors” (the analog of binary multiplication by a scalar λv) can be defined as a multiaction (kρ-place action) introduced in DUPLIJ [2012] ρkρ : Kkρ × V −→ V. (2.5) The set of all multiactions form a nρ-ary semigroup Sρ under composition. We can “normalize” the multiactions in a similar way, as multiplace representations DUPLIJ [2012], by (an analog of 1v = v, v ∈ V, 1 ∈ K) ρkρ    eK ... eK v    = v, (2.6) for all v ∈ V, where eK is the identity of KmK ,nK . In the case of an (ordinary) 1-place (left) action (as an external binary operation) ρ1 : K × V → V, its consistency with the polyadic field multiplication κnK under composition of the binary operations ρ1 {λ|a} gives a product of the same arity nρ = nK, (2.7)
6 STEVEN DUPLIJ that is (a polyadic analog of λ (µv) = (λµ) v, v ∈ V, λ, µ ∈ K) ρ1 {λ1|ρ1 {λ2| . . . |ρ1 {λnK |v}} . . .} = ρ1 {κnK [λ1, λ2, . . . λnK ] |v} , λ1, . . . , λn ∈ K, v ∈ V. (2.8) In the general case of kρ-place actions, the multiplication in the nρ-ary semigroup Sρ can be defined by the changing arity formula DUPLIJ [2012] (schematically) ρkρ nρ    λ1 ... λkρ . . . ρkρ    λkρ(nρ−1) ... λkρnρ v    . . .    = ρkρ    κnK [λ1, . . . λnK ] , ... κnK λnK(ℓµ−1), . . . λnKℓµ    ℓµ λnKℓµ+1, ... λnKℓµ+ℓid    ℓid v    , (2.9) where ℓµ and ℓid are both integers. The associativity of (2.9) in each concrete case can be achieved by applying the associativity quiver concept from DUPLIJ [2012]. Definition 2.1. The ℓ-shape is a pair (ℓµ, ℓid) , where ℓµ is the number of multiplications and ℓid is the number of intact elements in the composition of operations. It follows from (2.9), Proposition 2.2. The arities of the polyadic field KmK ,nK , the arity nρ of the multiaction semigroup Sρ and the ℓ-shape of the composition satisfy kρnρ = nKℓµ + ℓid, (2.10) kρ = ℓµ + ℓid. (2.11) We can exclude ℓµ or ℓid and obtain nρ = nK − nK − 1 kρ ℓid, nρ = nK − 1 kρ ℓµ + 1, (2.12) respectively, where nK−1 kρ ℓid ≥ 1 and nK−1 kρ ℓµ ≥ 1 are integers. The following inequalities hold 1 ≤ ℓµ ≤ kρ, 0 ≤ ℓid ≤ kρ − 1, ℓµ ≤ kρ ≤ (nK − 1) ℓµ, 2 ≤ nρ ≤ nK. (2.13) Remark 2.3. The formulas (2.12) coincide with the arity changing formulas for heteromorphisms DUPLIJ [2012] applied to (2.9). It follows from (2.10), that the ℓ-shape is determined by the arities and number of places kρ by ℓµ = kρ (nρ − 1) nK − 1 , ℓid = kρ (nK − nρ) nK − 1 . (2.14) Because we have two polyadic “additions” νmV and σmK , we need to consider, how the multiaction ρkρ “distributes” between each of them. First, consider distributivity of the multiaction ρkρ with respect to “vector addition” νmV (a polyadic analog of the binary λ (v + u) = λv + λu, v, u ∈ V, λ, µ ∈ K) ρkρ    λ1 ... λkρ νmV [v1, . . . , vmV ]    = νmV  ρkρ    λ1 ... λkρ v1    , . . . , ρkρ    λ1 ... λkρ vmV      . (2.15) Observe that here, in distinction to (2.9), there is no connection between the arities mV and kρ.
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 7 Secondly, the distributivity of the multiaction ρkρ (“multiplication by scalar”) with respect to the “field addition” (a polyadic analog of (λ + µ) v = λv + µv, v ∈ A, λ, µ ∈ K) has a form similar to (2.9) (which can be obtained from the changing arity formula DUPLIJ [2012]) ρkρ nρ    λ1 ... λkρ . . . ρkρ    λkρ(nρ−1) ... λkρnρ v    . . .    = ρkρ    σmK [λ1, . . . λmK ] , ... σmK λmK (ℓ′ µ−1), . . . λmK ℓ′ µ    ℓ′ µ λmK ℓ′ µ+1, ... λmK ℓ′ µ+ℓid    ℓ′ id v    , (2.16) where ℓ′ ρ and ℓ′ id are the numbers of multiplications and intact elements in the resulting multiaction, respectively. Here the arities are not independent as in (2.15), and so we have Proposition 2.4. The arities of the polyadic field KmK ,nK , the arity nρ of the multiaction semigroup Sρ and the ℓ-shape of the distributivity satisfy kρnρ = mKℓ′ µ + ℓ′ id, (2.17) kρ = ℓ′ µ + ℓ′ id. (2.18) It follows from (2.17)–(2.18) nρ = mK − mK − 1 kρ ℓ′ id, nρ = mK − 1 kρ ℓ′ µ + 1. (2.19) Here mK −1 kρ ℓ′ id ≥ 1 and mK −1 kρ ℓ′ µ ≥ 1 are integers, and we have the inequalities 1 ≤ ℓ′ µ ≤ kρ, 0 ≤ ℓ′ id ≤ kρ − 1, ℓ′ µ ≤ kρ ≤ (mK − 1) ℓ′ µ, 2 ≤ nρ ≤ mK. (2.20) Now, the ℓ-shape of the distributivity is fully determined from the arities and number of places kρ by the arity shape formulas ℓ′ ρ = kρ (nρ − 1) mK − 1 , ℓ′ id = kρ (mK − nρ) mK − 1 . (2.21) It follows from (2.20) that: Corollary 2.5. The arity nρ of the multiaction semigroup Sρ is less than or equal to the arity of the field addition mK. Definition 2.6. A polyadic (K)-vector (“linear”) space over a polyadic field is the 2-set 4-operation algebraic structure VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ , (2.22) such that the following axioms hold: 1) K | σmK , κnK is a polyadic (mK, nK)-field KmK ,nK ; 2) A | νmV is a commutative mV -ary group; 3) ρkρ | composition is a nρ-ary semigroup Sρ; 4) Distributivity of the multiaction ρkρ with respect to the “vector addition” νmV (2.15); 5) Distributivity of ρkρ with respect to the “scalar addition” σmK (2.16); 6) Compatibility of ρkρ with the “scalar multiplication” κnK (2.9); 7) Normalization of the multiaction ρkρ (2.6).
8 STEVEN DUPLIJ All of the arities in (2.22) are independent and can be chosen arbitrarily, but they fix the ℓ-shape of the multiaction composition (2.9) and the distributivity (2.16) by (2.14) and (2.21), respectively. Note that the main distinction from the binary case is the possibility for the arity nρ of the multiaction semigroup Sρ to be arbitrary. Definition 2.7. A polyadic K-vector subspace is Vsub mK ,nK,mV ,kρ = K; Vsub | σmK , κnK ; νmV | ρkρ , (2.23) where the subset Vsub ⊂ V is closed under all operations σmK , κnK , νmV , ρkρ and the axioms 1)-7). Let us consider a subset S = {v1, . . . , vdV } ⊆ V (of dV “vectors”), then a polyadic span of S is (a “linear combination”) Spanλ pol (v1, . . . , vdV ) = {w} , (2.24) w = νℓν mV  ρkρ    λ1 ... λkρ v1    , . . . , ρkρ    λ(dV −1)kρ ... λdV kρ vs      , (2.25) where (dV · kρ) “scalars” play the role of coefficients (or coordinates as columns consisting of kρ elements from the polyadic field KmK ,nK ), and the number of “vectors” s is connected with the “number of mV -ary additions” ℓν by dV = ℓν (mV − 1) + 1, (2.26) while Spanλ pol S is the set of all “vectors” of this form (2.24) (we consider here finite “sums” only). Definition 2.8. A polyadic span S = {v1, . . . , vdV } ⊆ V is nontrivial, if at least one multiaction ρkρ in (2.24) is nonzero. Since polyadic fields and groups may not contain zeroes, we need to redefine the basic notions of equivalences. Let us take two different spans of the same set S. Definition 2.9. A set {v1, . . . , vdV } is called “linear” polyadic independent, if from the equality of nontrivial spans, as Spanλ pol (v1, . . . , vdV ) = Spanλ′ pol (v1, . . . , vdV ), it follows that all λi = λ′ i, i = 1, . . . , dV kρ. Definition 2.10. A set {v1, . . . , vdV } is called a polyadic basis of a polyadic vector space VmK nK mV kρ , if it spans the whole space Spanλ pol (v1, . . . , vdV ) = V. In other words, any element of V can be uniquely presented in the form of the polyadic “linear combination” (2.24). If a polyadic vector space VmK nK mV kρ has a finite basis {v1, . . . , vdV }, then any another basis v′ 1, . . . , v′ dV has the same number of elements. Definition 2.11. The number of elements in the polyadic basis {v1, . . . , vdV } is called the polyadic dimension of VmK ,nK ,mV ,kρ. Remark 2.12. The so-called 3-vector space introduced and studied in DUPLIJ AND WERNER [2015], corresponds to VmK =3,nK=2,mV =3,kρ=1. 2.2. One-set polyadic vector space. A particular polyadic vector space is important: consider V = K, νmV = σmK and mV = mK, which gives the following one-set “linear” algebraic structure (we call it a one-set polyadic vector space) KmK ,nK,kρ = K | σmK , κnK | ρλ kρ , (2.27)
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 9 where now the multiaction ρλ kρ    λ1 ... λkρ λ    , λ, λi ∈ K, (2.28) acts on K itself (in some special way), and therefore can be called a regular multiaction. In the binary case nK = mK = 2, the only possibility for the regular action is the multiplication (by “scalars”) in the field ρλ 1 {λ1| λ} = κ2 [λ1λ] (≡ λ1λ), which obviously satisfies the axioms 4)-7) of a vector space in Definition 2.6. In this way we arrive at the definition of the binary field K ≡ K2,2 = K | σ2, κ2 , and so a one-set binary vector space coincides with the underlying field KmK =2,nK=2,kρ=1 = K, or as it is said “a field is a (one-dimensional) vector space over itself”. Remark 2.13. In the polyadic case, the regular multiaction ρλ kρ can be chosen, as any (additional to σmK , κnK ) function satisfying axioms 4)-7) of a polyadic vector space and the number of places kρ and the arity of the semigroup of multiactions Sρ can be arbitrary, in general. Also, ρλ kρ can be taken as a some nontrivial combination of σmK , κnK satisfying axioms 4)-7) (which admits a nontrivial “multiplication by scalars”). In the simplest regular (similar to the binary) case, ρλ kρ    λ1 ... λkρ λ    = κℓκ nK λ1, . . . , λkρ, λ , (2.29) where ℓκ is the number of multiplications κnK , and the number of places kρ is now fixed by kρ = ℓκ (nK − 1) , (2.30) while λ in (2.29) can be on any place due the commutativity of the field multiplication κnK . Remark 2.14. In general, the one-set polyadic vector space need not coincide with the underlying polyadic field, KmK ,nK ,kρ = KnK mK (as opposed to the binary case), but can have a more complicated structure which is determined by an additional multiplace function, the multiaction ρλ kρ . 2.3. Polyadic algebras. By analogy with the binary case, introducing an additional operation on vectors, a multiplication which is distributive and “linear” with respect to “scalars”, leads to a polyadic generalization of the (associative) algebra notion CARLSSON [1980]. Here, we denote the second (except for the ’scalars’ K) set by A (instead of V above), on which we define two operations: mA-ary “addition” νmA : A×mA → A and nA-ary “multiplication” µnA : A×nA → A. To interpret the nA-ary operation as a true multiplication, the operations µnA and νmA should satisfy polyadic distributivity (1.1)–(1.3) (an analog of (a + b) · c = a · c + b · c, with a, b, c ∈ A). Then we should consider the “interaction” of this new operation µnA with the multiaction ρkρ (an analog of the “compatibility with scalars” (λa) · (µb) = (λµ) a · b, a, b ∈ A, λ, µ ∈ K). In the most general case, when all arities are
10 STEVEN DUPLIJ arbitrary, we have the polyadic compatibility of µnA with the field multiplication κnK as follows µnA  ρkρ    λ1 ... λkρ a1    , . . . , ρkρ    λkρ(nA−1) ... λkρnA anA      = ρkρ    κnK [λ1, . . . , λnK ] , ... κnK λnK (ℓ′′ µ−1), . . . , λnKℓ′′ µ    ℓ′′ µ λnKℓ′′ µ+1, ... λnKℓ′′ µ+ℓ′′ id    ℓ′′ id µnA [a1 . . . anA ]    , (2.31) where ℓ′′ µ and ℓ′′ id are the numbers of multiplications and intact elements in the resulting multiaction, respectively. Proposition 2.15. The arities of the polyadic field KmK ,nK , the arity nρ of the multiaction semigroup Sρ and the ℓ-shape of the polyadic compatibility (2.31) satisfy kρnA = nKℓ′′ µ + ℓ′′ id, kρ = ℓ′′ µ + ℓ′′ id. (2.32) We can exclude from (2.32) ℓ′′ ρ or ℓ′′ id and obtain nA = nK − nK − 1 kρ ℓ′′ id, nA = nK − 1 kρ ℓ′′ µ + 1, (2.33) where nK −1 kρ ℓ′′ id ≥ 1 and nK−1 kρ ℓ′′ µ ≥ 1 are integer, and the inequalities hold 1 ≤ ℓ′′ µ ≤ kρ, 0 ≤ ℓ′′ id ≤ kρ − 1, ℓ′′ µ ≤ kρ ≤ (nK − 1) ℓ′′ µ, 2 ≤ nA ≤ nK. (2.34) It follows from (2.32), that the ℓ-shape is determined by the arities and number of places kρ as ℓ′′ µ = kρ (nA − 1) nK − 1 , ℓ′′ id = kρ (nK − nA) nK − 1 . (2.35) Definition 2.16. A polyadic (“linear”) algebra over a polyadic field is the 2-set 5-operation algebraic structure AmK ,nK,mA,nA,kρ = K; A | σmK , κnK ; νmA , µnA | ρkρ , (2.36) such that the following axioms hold: 1) K; A | σmK , κnK ; νmA | ρkρ is a polyadic vector space over a polyadic field KmK ,nK ; 2) The algebra multiplication µnA and the algebra addition νmA satisfy the polyadic distributivity (1.1)–(1.3); 3) The multiplications in the algebra µnA and in the field κnK are compatible by (2.31). In the case where the algebra multiplication µnA is associative (1.4), then AmK,nK ,mA,nA,kρ is an associative polyadic algebra (for kρ = 1 see CARLSSON [1980]). If µnA is commutative, µnA [aA] = µnA [σ ◦ aA], for any polyad in algebra aA ∈ A×nA for all permutations σ ∈ Sn, where Sn is the symmetry group, then AmK ,nK,mA,nA,kρ is called a commutative polyadic algebra. As in the n-ary (semi)group theory, for polyadic algebras one can introduce special kinds of associativity and partial commutativity. If the multiplication µnA contains the identity eA (1.6) or a neutral polyad for any element, such a polyadic algebra is called unital or neutral-unital, respectively. It follows from (2.34) that: Corollary 2.17. In a polyadic (“linear”) algebra the arity of the algebra multiplication nA is less than or equal to the arity of the field multiplication nK.
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 11 Proposition 2.18. It all the operation ℓ-shapes in (2.9), (2.16) and (2.31) coincide ℓ′′ µ = ℓ′ µ = ℓµ, ℓ′′ id = ℓ′ id = ℓid, (2.37) then, we obtain the conditions for the arities nK = mK, nρ = nA, (2.38) while mA and kρ are not connected. Proof. Use (2.14) and (2.35). Proposition 2.19. In the case of equal ℓ-shapes the multiplication and addition of the ground polyadic field (“scalars”) should coincide, while the arity nρ of the multiaction semigroup Sρ should be the same as of the algebra multiplication nA, while the arity of the algebra addition mA and number of places kρ remain arbitrary. Remark 2.20. The above ℓ-shapes (2.14), (2.21), and (2.35) are defined by a pair of integers, and therefore the arities in them are not arbitrary, but should be “quantized” in the same manner as the arities of heteromorphisms in DUPLIJ [2012]. Therefore, numerically the “quantization” rules for the ℓ-shapes (2.14), (2.21), and (2.35) coincide and given in TABLE 1. TABLE 1. “Quantization” of arity ℓ-shapes kρ ℓµ | ℓ′ µ | ℓ′′ µ ℓid | ℓ′ id | ℓ′′ id nK nρ | mK nρ | nK nA 2 1 1 3, 5, 7, . . . 2, 3, 4, . . . 3 1 2 4, 7, 10, . . . 2, 3, 4, . . . 3 2 1 4, 7, 10, . . . 3, 5, 7, . . . 4 1 3 5, 9, 13, . . . 2, 3, 4, . . . 4 2 2 3, 5, 7, . . . 2, 3, 4, . . . 4 3 1 5, 9, 13, . . . 4, 7, 10, . . . Thus, we arrive at the following Theorem 2.21 (The arity partial freedom principle). The basic two-set polyadic algebraic structures have non-free underlying operation arities which are “quantized” in such a way that their ℓ-shape is given by integers. The above definitions can be generalized, as in the binary case by considering a polyadic ring RmK ,nK instead of a polyadic field KmK ,nK . In this way a polyadic vector space becomes a polyadic module over a ring or polyadic R-module, while a polyadic algebra over a polyadic field becomes a polyadic algebra over a ring or polyadic R-algebra. All the axioms and relations between arities in the Definition 2.6 and Definition 2.16 remain the same. However, one should take into account that the ring multiplication κnK can be noncommutative, and therefore for polyadic R-modules and R- algebras it is necessary to consider many different kinds of multiactions ρkρ (all of them are described in (2.9)). For instance, in the ternary case this corresponds to left, right and central ternary modules, and tri-modules CARLSSON [1976], BAZUNOVA ET AL. [2004].
12 STEVEN DUPLIJ 3. MAPPINGS BETWEEN POLYADIC ALGEBRAIC STRUCTURES Let us consider DV different polyadic vector spaces over the same polyadic field KmK ,nK , as V (i) mK ,nK,m (i) V ,k (i) ρ = K; V(i) | σmK , κnK ; ν (i) m (i) V | ρ (i) k (i) ρ , i = 1, . . . , DV < ∞. (3.1) Here we define a polyadic analog of a “linear” mapping for polyadic vector spaces which “com- mutes“ with the “vector addition” and the “multiplication by scalar” (an analog of the additivity F (v + u) = F (v) + F (u), and the homogeneity of degree one F (λv) = λF (v), v, u ∈ V, λ ∈ K). Definition 3.1. A 1-place (“K-linear”) mapping between polyadic vector spaces VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ and VmK ,nK,mV ,kρ = K; V′ | σmK , κnK ; ν′ mV | ρ′ kρ over the same polyadic field KmK ,nK = K | σmK , κnK is F1 : V → V′ , such that F1 (νmV [v1, . . . , vmV ]) = ν′ mV [F1 (v1) , . . . , F1 (vmV )] , (3.2) F1  ρkρ    λ1 ... λkρ v      = ρ′ kρ    λ1 ... λkρ F1 (v)    , (3.3) where v1, . . . , vmV , v ∈ V, λ1, . . . , λkρ ∈ K. If zV is a “zero vector” in V and zV ′ is a “zero vector” in V′ (see (2.2)), then it follows from (3.2)–(3.3), that F1 (zV ) = zV ′ . The initial and final arities of νmV (“vector addition”) and the multiaction ρkρ (“multiplication by scalar”) coincide, because F1 is a 1-place mapping (a linear homomorphism). In DUPLIJ [2012] multiplace mappings and corresponding heteromorphisms were introduced. The latter allow us to change arities (mV → m′ V , kρ → k′ ρ), which is the main difference between binary and polyadic mappings. Definition 3.2. A kF -place (“K-linear”) mapping between two polyadic vector spaces VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ and VmK ,nK,mV ,kρ = K; V′ | σmK , κnK ; ν′ m′ V | ρ′ k′ ρ over the same polyadic field KmK ,nK = K | σmK , κnK is defined, if there exists FkF : V×kF → V′ , such that FkF            νmV [v1, . . . , vmV ] ... νmV vmV (ℓk µ−1), . . . vmV ℓk µ    ℓk µ vmV ℓk µ+1, ... vmV ℓk µ+ℓk id    ℓk id            = ν′ m′ V   FkF   v1 ... vkF   , . . . , FkF    vkF (m′ V −1) ... vkF m′ V       , (3.4)
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 13 FkF                    ρkρ    λ1 ... λkρ v1    ... ρkρ    λkρ(ℓf µ−1) ... λkρℓf µ vℓf µ       ℓf µ vℓf µ+1 ... vkF    ℓf id                    = ρ′ k′ ρ    λ1 ... λk′ ρ FkF   v1 ... vkF      , (3.5) where v1, . . . , vmV , v ∈ V, λ1, . . . , λkρ ∈ K, and the four integers ℓk ρ, ℓk id, ℓf ρ, ℓf id define the ℓ-shape of the mapping. It follows from (3.4)–(3.5), that the arities satisfy kF m′ V = mV ℓk µ + ℓk id, kF = ℓk µ + ℓk id, kF = ℓf µ + ℓf id, k′ µ = kρℓf µ. (3.6) The following inequalities hold 1 ≤ ℓk µ ≤ kF , 0 ≤ ℓk id ≤ kF − 1, ℓk µ ≤ kF ≤ (mV − 1) ℓk µ, 2 ≤ m′ V ≤ mV , 2 ≤ kρ ≤ k′ ρ. (3.7) Thus, the ℓ-shape of the kF -place mapping between polyadic vector spaces is determined by ℓk µ = kF (mV − 1) mV − 1 , ℓk id = kF (mV − m′ V ) mV − 1 , ℓf µ = kρ k′ ρ , ℓf id = kF − kρ k′ ρ . (3.8) 3.1. Polyadic functionals and dual polyadic vector spaces. An important particular case of the kF - place mapping can be considered, where the final polyadic vector space coincides with the underlying field (analog of a “linear functional”). Definition 3.3. A “linear” polyadic functional (or polyadic dual vector, polyadic covector) is a kL- place mapping of a polyadic vector space VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ into its polyadic field KmK ,nK = K | σmK , κnK , such that there exists LkL : V×kL → K, and LkL            νmV [v1, . . . , vmV ] ... νmV vmV (ℓk ν−1), . . . vmV ℓk ν    ℓk ν vmV ℓk ν+1, ... vnK ℓk ν+ℓν id    ℓν id            = σmK  LkL   v1 ... vkL   , . . . , LkL   vkL(mK −1) ... vkLmK     , (3.9)
14 STEVEN DUPLIJ LkL                    ρkρ    λ1 ... λkρ v1    ... ρkρ    λkρ(ℓL µ −1) ... λkρℓL µ vℓL µ       ℓL µ vℓL µ +1 ... vkL    ℓL id                    = κnK  λ1, . . . , λnK−1, LkL   v1 ... vkL     , (3.10) where v1, . . . , vmV , v ∈ V , λ1, . . . , λnK ∈ K, and the integers ℓk ν, ℓν id, ℓL µ, ℓL id define the ℓ-shape of LkL . It follows from (3.4)–(3.5), that the arities satisfy kLmK = mV ℓk ν + ℓν id, kL = ℓk ν + ℓν id, kL = ℓh µ + ℓh id, nK − 1 = kρℓh µ, (3.11) and for them 1 ≤ ℓk ν ≤ kL, 0 ≤ ℓν id ≤ kL −1, ℓk ν ≤ kL ≤ (mV − 1) ℓk ν, 2 ≤ mK ≤ mV , 2 ≤ kρ ≤ nK −1. (3.12) Thus, the ℓ-shape of the polyadic functional is determined by ℓk ν = kL (mK − 1) mV − 1 , ℓν id = kL (mV − mK) mV − 1 , ℓh µ = kρ nK − 1 , ℓh id = kL − kρ nK − 1 . (3.13) In the binary case, because the dual vectors (linear functionals) take their values in the underlying field, new operations between them, such that the dual vector “addition” (+∗ ) and the “multiplication by a scalar” (•∗ ) can be naturally introduced by L(1) +∗ L(2) (v) = L(1) (v)+L(2) (v), (λ •∗ L) (v) = λ•L (v), which leads to another vector space structure, called a dual vector space. Note that operations +∗ and +, •∗ and • are different, because + and • are the operations in the underlying field K. In the polyadic case, we have more complicated arity changing formulas, and here we consider finite- dimensional spaces only. The arities of operations between dual vectors can be different from ones in the underlying polyadic field KmK nK , in general. In this way, we arrive to the following Definition 3.4. A polyadic dual vector space over a polyadic field KmK ,nK is V∗ mK ,nK,m∗ V ,k∗ ρ = K; L (i) kL | σmK , κnK ; ν∗ mL | ρ∗ kL , (3.14) and the axioms are: 1) K | σmK , κnK is a polyadic (mK, nK)-field KmK ,nK ; 2) L (i) kL | ν∗ mL , i = 1, . . . , DL is a commutative mL-ary group (which is finite, if DL < ∞); 3) The “dual vector addition” ν∗ mL is compatible with the polyadic field addition σmK by ν∗ mL L (1) kL , . . . , L (mL) kL a(kL) = σmK L (1) kL a(kL) , . . . , L (mK ) kL v(kL) , (3.15) where v(kL) =   v1 ... vkL  , v1, . . . , vkL ∈ V, and it follows that mL = mK. (3.16)
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 15 4) The compatibility of ρ∗ kL with the “multiplication by a scalar” in the underlying polyadic field ρ∗ kL    λ1 ... λkL LkL    v(kL) = κnK λ1, . . . , λnK−1, LkL v(kL) , (3.17) and then kL = nK − 1 (3.18) 5) ρ∗ kL | composition is a nL-ary semigroup SL (similar to (2.9)) ρ∗ kL nL    λ1 ... λkL . . . ρ∗ kL    λkL(nL−1) ... λkLnL LkL    . . .    v(kL) (3.19) = ρ∗ kL    κnK [λ1, . . . λnK ] , ... κnK λnK(ℓL µ −1), . . . λnK ℓL µ    ℓL µ λnK ℓL µ +1, ... λnKℓL µ +ℓL id    ℓL id LkL    v(kL) , (3.20) where the ℓ-shape is determined by the system kLnL = nKℓL µ + ℓL id, kL = ℓL µ + ℓL id. (3.21) Using (3.18) and (3.21), we obtain the ℓ-shape as ℓL µ = nL − 1, ℓL id = nK − nL. (3.22) Corollary 3.5. The arity nL of the semigroup SL is less than or equal to the arity nK of the underlying polyadic field nL ≤ nK. 3.2. Polyadic direct sum and tensor product. The Cartesian product of DV polyadic vector spaces ×ΠmV i=1V (i) mK nKm (i) V k (i) ρ (sometimes we use the concise notation ×ΠV(i) ), i = 1, . . . , DV is given by the DV -ples (an analog of the Cartesian pair (v, u), v ∈ V(1) , u ∈ V(2) )    v(1) ... v(DV )    ≡ v(DV ) ∈ V×DV . (3.23) We introduce a polyadic generalization of the direct sum and tensor product of vector spaces by considering “linear” operations on the DV -ples (3.23). In the first case, to endow ×ΠV(i) with the structure of a vector space we need to define a new operation between the DV -ples (3.23) (similar to vector addition, but between elements from different spaces) and a rule, specifying how they are “multiplied by scalars” (analogs of (v1, v2) + (u1, u2) = (v1 + u1, v2 + u2) and λ (v1, v2) = (λv1, λv2) ). In the binary case, a formal summation is used, but it can be different from the addition in the initial vector spaces. Therefore, we can define on the set of the DV -ples (3.23) new operations χmV (“addition of vectors from different spaces”) and “multiplication by a scalar” τkρ , which does not need to coincide with the corresponding operations ν (i) m (i) V and ρ (i) k (i) ρ of the initial polyadic vector spaces V (i) mK ,nK,m (i) V ,k (i) ρ .
16 STEVEN DUPLIJ If all DV -ples (3.23) are of fixed length, then we can define their “addition” χmV in the standard way, if all the arities m (i) V coincide and equal the arity of the resulting vector space mV = m (1) V = . . . = m (DV ) V , (3.24) while the operations (“additions”) themselves ν (i) mV between vectors in different spaces can be still different. Thus, a new commutative mV -ary operation (“addition”) χmV of the DV -ples of the same length is defined by χmV       v (1) 1 ... v (DV ) 1    , . . . ,    v (1) mV ... v (DV ) mV       =      ν (1) mV v (1) 1 , . . . , v (1) mV ... ν (DV ) mV v (DV ) 1 , . . . , v (DV ) mV      , (3.25) where DV = mV , in general. However, it is also possible to add DV -ples of different length such that the operation (3.25) is compatible with all arities m (i) V , i = 1, . . . , mV . For instance, if mV = 3, m (1) V = m (2) V = 3, m (3) V = 2, then χ3       v (1) 1 v (2) 1 v (3) 1    ,    v (1) 2 v (2) 2 v (3) 2    ,   v (1) 3 v (2) 3      =      ν (1) 3 v (1) 1 , v (1) 2 , v (1) 3 ν (2) 3 v (2) 1 , v (2) 2 , v (2) 3 ν (3) 2 v (3) 1 , v (3) 2      . (3.26) Assertion 3.6. In the polyadic case, a direct sum of polyadic vector spaces having different arities of “vector addition” m (i) V can be defined. Let us introduce the multiaction τkρ (“multiplication by a scalar”) acting on DV -ple v(mV ) , then ρkρ    λ1 ... λkρ    v(1) ... v(DV )       =               ρ (1) k (1) ρ    λ1 ... λk (1) ρ v(1)    ... ρ (mV ) k (DV ) ρ    λ k (1) ρ +...+k (DV −1) ρ +1 ... λ k (1) ρ +...+k (DV ) ρ v(DV )                  , (3.27) where k(1) ρ + . . . + k(DV ) ρ = kρ. (3.28) Definition 3.7. A polyadic direct sum of mV polyadic vector spaces is their Cartesian product equipped with the mV -ary addition χmV and the kρ-place multiaction τkρ , satisfying (3.25) and (3.27) respectively ⊕ ΠDV i=1V (i) mK ,nK,m (i) V ,k (i) ρ = ×ΠDV i=1V (i) mK ,nK,m (i) V ,k (i) ρ | χmV , τkρ . (3.29) Let us consider another way to define a vector space structure on the DV -ples from the Carte- sian product ×ΠV(i) . Remember that in the binary case, the concept of bilinearity is used, which means “distributivity” and “multiplicativity by scalars” on each place separately in the Cartesian pair
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 17 (v1, v2) ∈ V(1) × V(2) (as opposed to the direct sum, where these relations hold on all places simulta- neously, see (3.25) and (3.27)) such that (v1 + u1, v2) = (v1, v2) + (u1, v2) , (v1, v2 + u2) = (v1, v2) + (v1, u2) , (3.30) λ (v1, v2) = (λv1, v2) = (v1, λv2) , (3.31) respectively. If we denote the ideal corresponding to the relations (3.30)–(3.31) by B2, then the binary tensor product of the vector spaces can be defined as their Cartesian product by factoring out this ideal, as V(1) ⊗V(2) def = V(1) ×V(2) B2. Note first, that the additions and multiplications by a scalar on both sides of (3.30)–(3.31) “work” in different spaces, which sometimes can be concealed by adding the word “formal” to them. Second, all these operations have the same arity (binary ones), which need not be the case when considering polyadic structures. As in the case of the polyadic direct sum, we first define a new operation ˜χmV (“addition”) of the DV -ples of fixed length (different from χmV in (3.25)), when all the arities m (i) V coincide and equal to mV (3.24). Then, a straightforward generalization of (3.30) can be defined for mV -ples similar to the polyadic distributivity (1.1)–(1.3), as in the following mV relations      ν (1) mV v (1) 1 , . . . , v (1) mV u2 ... uDV      = ˜χmV           v (1) 1 u2 ... uDV      , . . . ,      v (1) mV u2 ... uDV           , (3.32)      u1 ν (2) mV v (2) 1 , . . . , v (2) mV ... umV      = ˜χmV           u1 v (2) 1 ... umV      , . . . ,      u1 v (2) mV ... umV           , (3.33) ...      u1 u2 ... ν (mV ) mV v (DV ) 1 , . . . , v (DV ) mV      = ˜χmV           u1 u2 ... v (DV ) 1      , . . . ,      u1 u2 ... v (DV ) mV           . (3.34) By analogy, if all k (i) ρ are equal we can define a new multiaction ˜τkρ (different from τkρ (3.27)) with the same number of places kρ = k(1) ρ = . . . = k(DV ) ρ (3.35) as the DV relations (an analog of (3.31)) ρ′ kρ    λ1 ... λkρ    v(1) ... v(DV )       =          ρ (1) kρ    λ1 ... λkρ v(1)    v(2) ... v(DV )          (3.36)
18 STEVEN DUPLIJ =          v(1) ρ (2) kρ    λ1 ... λkρ v(2)    ... v(DV )          (3.37) ... =          v(1) v(2) ... ρ (DV ) kρ    λ1 ... λkρ v(DV )             . (3.38) Let us denote the ideal corresponding to the relations (3.32)–(3.34), (3.36)–(3.38) by BDV . Definition 3.8. A polyadic tensor product of DV polyadic vector spaces V (i) mK ,nK,m (i) V ,k (i) ρ is their Carte- sian product equipped with the mV -ary addition ˜χmV (of DV -ples) and the kρ-place multiaction ˜τkρ , satisfying (3.32)–(3.34) and (3.36)–(3.38), respectively, by factoring out the ideal BDV ⊗ ΠmV i=1V (i) mK ,nK,m (i) V ,k (i) ρ = ×ΠmV i=1V (i) mK ,nK,m (i) V ,k (i) ρ | ˜χmV , ˜τkρ BDV . (3.39) As in the case of the polyadic direct sum, we can consider the distributivity for DV -ples of different length. In a similar example (3.26), if mV = 3, m (1) V = m (2) V = 3, m (3) V = 2, we have    ν (1) 3 v (1) 1 , v (1) 2 , v (1) 3 u2 u3    = ˜χ3     v (1) 1 u2 u3   ,   v (1) 2 u2 u3   ,   v (1) 3 u2     , (3.40)    u1 ν (2) 3 v (2) 1 , v (2) 2 , v (2) 3 u3    = ˜χ3     u1 v (2) 1 u3   ,   u1 v (2) 2 u3   ,   u1 v (2) 3     , (3.41)    u1 u2 ν (3) 2 v (3) 1 , v (3) 2    = ˜χ3     u1 u2 v (3) 1   ,   u1 u2 v (3) 2   ,   u1 u2     . (3.42) Assertion 3.9. A tensor product of polyadic vector spaces having different arities of the “vector addition” m (i) V can be defined. In the case of modules over a polyadic ring, the formulas connecting arities and ℓ-shapes similar to those above hold, while concrete properties (noncommutativity, mediality, etc.) should be taken into account.
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 19 4. POLYADIC INNER PAIRING SPACES AND NORMS Here we introduce the next important operation: a polyadic analog of the inner product for polyadic vector spaces - a polyadic inner pairing2 . Let VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ be a polyadic vector space over the polyadic field KmK ,nK (2.22). By analogy with the binary inner product, we introduce its polyadic counterpart and study its arity shape. Definition 4.1. A polyadic N-place inner pairing (an analog of the inner product) is a mapping N •|•| . . . |• : V×N → K, (4.1) satisfying the following conditions: 1) Polyadic “linearity” (2.9) (for first argument): ρkρ    λ1 ... λkρ v1    |v2| . . . |vN = κnK λ1, . . . , λkρ , v1|v2| . . . |vN . (4.2) 2) Polyadic “distributivity” (1.1)–(1.3) (on each place): νmV [v1, u1, . . . umV −1] |v2| . . . |vN = σmK [ v1|v2| . . . |vN , u1|v2| . . . |vN . . . umV −1|v2| . . . |vN ] . (4.3) If the polyadic field KmK ,nK contains the zero zK and V | mV has the zero “vector” zV (which is not always the case in the polyadic case), we have the additional axiom: 3) The polyadic inner pairing vanishes v1|v2| . . . |vN = zK, iff any of the “vectors” vanishes, ∃i ∈ 1, . . . , N, such that vi = zV . If the standard binary ordering on KmK ,nK can be defined, then the polyadic inner pairing satisfies: 4) The positivity condition N v|v| . . . |v ≥ zK, (4.4) 5) The polyadic Cauchy-Schwarz inequality (“triangle” inequality) κnK       nK N v1|v1| . . . |v1 , N v2|v2| . . . |v2 . . . N vnK |vnK | . . . |vnK       ≥ κnK   nK v1|v2| . . . |vN , v1|v2| . . . |vN . . . , v1|v2| . . . |vN   . (4.5) To make the above relations consistent, the arity shapes should be fixed. Definition 4.2. If the inner pairing is fully symmetric under permutations it is called a polyadic inner product. Proposition 4.3. The number of places in the multiaction ρkρ differs by 1 from the multiplication arity of the polyadic field nK − kρ = 1. (4.6) Proof. It follows from the polyadic “linearity” (4.2). 2 Note that this concept is different from the n-inner product spaces considered in [Misiak, et al].
20 STEVEN DUPLIJ Proposition 4.4. The arities of “vector addition” and “field addition” coincide mV = mK. (4.7) Proof. Implied by the polyadic “distributivity” (4.3). Proposition 4.5. The arity of the “field multiplication” is equal to the arity of the polyadic inner pairing space nK = N. (4.8) Proof. This follows from the polyadic Cauchy-Schwarz inequality (4.5). Definition 4.6. The polyadic vector space VmK ,nK,mV ,kρ equipped with the polyadic inner pairing N •|•| . . . |• : V×N → K is called a polyadic inner pairing space HmK ,nK,mV ,kρ,N . A polyadic analog of the binary norm • : V → K can be induced by the inner pairing similarly to the binary case for the inner product (we use the form v 2 = v|v ). Definition 4.7. A polyadic norm of a “vector” v in the polyadic inner pairing space HmK ,nK,mV ,kρ,N is a mapping • N : V → K, such that κnK   nK v N , v N , . . . , v N   = N v|v| . . . |v , (4.9) nK = N, (4.10) and the following axioms apply: 1) The polyadic “linearity” ρkρ    λ1 ... λkρ v    N = κnK λ1, . . . , λkρ , v N , (4.11) nK − kρ = 1. (4.12) If the polyadic field KmK ,nK contains the zero zK and V | mV has a zero “vector” zV , then: 2) The polyadic norm vanishes v N = zK, iff v = zV . If the binary ordering on V | mV can be defined, then: 3) The polyadic norm is positive v N ≥ zK. 4) The polyadic“triangle” inequality holds σmK   mK v1 N , v2 N , . . . , vN N   ≥ νmV   mV v1 N , v2 N , . . . , vN N   , (4.13) mK = mV = N. (4.14) Definition 4.8. The polyadic inner pairing space HmK ,nK ,mV ,kρ,N equipped with the polyadic norm v N is called a polyadic normed space. Recall that in the binary vector space V over the field K equipped with the inner product •|• and the norm • , one can introduce the angle between vectors v1 · v2 · cos θ = v1|v2 , where on l.h.s. there are two binary multiplications (·). Definition 4.9. A polyadic angle between N vectors v1, v2, . . . , vnK of the polyadic inner pairing space HmK ,nK,mV ,kρ,N is defined as a set of angles ϑ = {{θi} | i = 1, 2, . . . , nK − 1} satisfying κ(2) nK [ v1 N , v2 N , . . . , vnK N , cos θ1, cos θ2, . . . , cos θnK −1] = v1|v2| . . . |vnK , (4.15)
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 21 where κ (2) nK is a long product of two nK-ary multiplications, which consists of 2 (nK − 1) + 1 terms. We will not consider the completion with respect to the above norm (to obtain a polyadic analog of Hilbert space) and corresponding limits and boundedness questions, because it will not give additional arity shapes, in which we are mostly interested here. Instead, below we turn to some applications and new general constructions which appear from the above polyadic structures. TABLE 2. The arity signature and arity shape of polyadic algebraic structures. Structures Sets Operations and arities Arity N Name N Multiplications Additions Multiactions shape Group-like polyadic algebraic structures n-ary magma (or groupoid) 1 M 1 µn : Mn → M n-ary semigroup (and monoid) 1 S 1 µn : Sn → S n-ary quasigroup (and loop) 1 Q 1 µn : Qn → Q n-ary group 1 G 1 µn : Gn → G Ring-like polyadic algebraic structures (m, n)-ary ring 1 R 2 µn : Rn → R νm : Rm → R (m, n)-ary field 1 K 2 µn : Kn → K νm : Km → K Module-like polyadic algebraic structures Module over (m, n) -ring 2 R, M 4 σn : Rn → R κm : Rm → R νmM : MmM → M ρkρ : Rkρ × M → M Vector space over (mK , nK) -field 2 K, V 4 σnK : KnK → K κmK : KmK → K νmV : VmV → V ρkρ : Kkρ × V → V (2.14) (2.21) Algebra-like polyadic algebraic structures Inner pairing space over (mK , nK) -field 2 K, V 5 σnK : KnK → K N-Form •..• : VN → K κmK : KmK → K νmV : VmV → V ρkρ : Kkρ × V → V (4.6) (4.7) (4.8) (mA, nA) -algebra over (mK , nK) -field 2 K, A 5 σnK : KnK → K µnA : An → A κmK : KmK → K νmA : AmM → A ρkρ : Kkρ × A → A (2.35) To conclude, we present the resulting TABLE 2 in which the polyadic algebraic structures are listed together with their arity shapes. APPLICATIONS 5. ELEMENTS OF POLYADIC OPERATOR THEORY Here we consider the 1-place polyadic operators T = FkF =1 (the case kF = 1 of the mapping FkF in Definition 3.2) on polyadic inner pairing spaces and structurally generalize the adjointness and involution concepts. Remark 5.1. A polyadic operator is a complicated mapping between polyadic vector spaces having nontrivial arity shape (3.4) which is actually an action on a set of “vectors”. However, only for kF = 1 it can be written in a formal way multiplicatively, as it is always done in the binary case. Recall (to fix notations and observe analogies) the informal standard introduction of the operator algebra and the adjoint operator on a binary pre-Hilbert space H (≡ HmK =2,nK=2,mV =2,kρ=1,N=2) over a binary field K (≡ KmK =2,nK=2) (having the underlying set {K; V}). For the operator norm • T : {T } → K we use (among many others) the following definition T T = inf {M ∈ K | T v ≤ M v , ∀v ∈ V} , (5.1)
22 STEVEN DUPLIJ which is convenient for further polyadic generalization. Bounded operators have M < ∞. If on the set of operators {T } (as 1-place mappings V → V) one defines the addition (+T ), product (◦T ) and scalar multiplication (·T ) in the standard way (T1 +T T2) (v) = T1v + T2v, (5.2) (T1 ◦T T2) (v) = T1 (T2v) , (5.3) (λ ·T T ) (v) = λ (T v) , λ ∈ K, v ∈ V, (5.4) then {T } | +T , ◦T |·T becomes an operator algebra AT (associativity and distributivity are obvious). The unity I and zero Z of AT (if they exist), satisfy Iv = v, (5.5) Zv = zV , ∀v ∈ V, (5.6) respectively, where zV ∈ V is the polyadic “zero-vector”. The connection between operators, linear functionals and inner products is given by the Riesz rep- resentation theorem. Informally, it states that in a binary pre-Hilbert space H = {K; V} a (bounded) linear functional (sesquilinear form) L : V × V → K can be uniquely represented as L (v1, v2) = T v1|v2 sym , ∀v1, v2 ∈ V, (5.7) where •|• sym : V × V → K is a (binary) inner product with standard properties and T : V → V is a bounded linear operator, such that the norms of L and T coincide. Because the linear functionals form a dual space (see Subsection 3.1), the relation (5.7) fixes the shape of its elements. The main consequence of the Riesz representation theorem is the existence of the adjoint: for any (bounded) linear operator T : V → V there exists a (unique bounded) adjoint operator T ∗ : V → V satisfying L (v1, v2) = T v1|v2 sym = v1|T ∗ v2 sym , ∀v1, v2 ∈ V, (5.8) and the norms of T and T ∗ are equal. It follows from the conjugation symmetry of the standard binary inner product, that (5.8) coincides with v1|T v2 sym = T ∗ v1|v2 sym , ∀v1, v2 ∈ V. (5.9) However, when •|• has no symmetry (permutation, conjugation, etc., see, e.g. MIGNOT [1976]), it becomes the binary (N = 2) inner pairing (4.1), the binary adjoint consists of 2 opera- tors (T ⋆12 ) = (T ⋆21 ), T ⋆ij : V → V, which should be defined by 2 equations T v1|v2 = v1|T ⋆12 v2 , (5.10) v1|T v2 = T ⋆21 v1|v2 , (5.11) where (⋆12) = (⋆21) are 2 different star operations satisfying 2 relations T ⋆12⋆21 = T , (5.12) T ⋆21⋆12 = T. (5.13) If •|• = •|• sym is symmetric, it becomes the inner product in the pre-Hilbert space H and the equations (5.12)–(5.13) coincide, while the operation (∗) = (⋆12) = (⋆21) stands for the standard involution T ∗∗ = T . (5.14)
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 23 5.1. Multistars and polyadic adjoints. Consider now a special case of the polyadic inner pairing space (see Definition 4.6) HmK ,nK,mV ,kρ=1,N = K; V | σmK , κnK ; νmV | ρkρ=1 | N •| . . . |• (5.15) with 1-place multiaction ρkρ=1. Definition 5.2. The set of 1-place operators T : V → V together with the set of “scalars” K be- come a polyadic operator algebra AT = K; {T } | σmK , κnK ; ηmT , ωnT | θkF =1 , if the operations ηmT , ωnT , θkF =1 to define by ηmT [T1, T2, . . . , TmT ] (v) = νmV [T1v, T2v, . . . , TmT v] , (5.16) ωnT [T1, T2, . . . , TnT ] (v) = T1 (T2 . . . (TnT v)) , (5.17) θkF =1 {λ | T } (v) = ρkρ=1 {λ | T v} , ∀λ ∈ K, ∀v ∈ V. (5.18) The arity shape is fixed by Proposition 5.3. In the polyadic algebra AT the arity of the operator addition mT coincides with the “vector” addition of the inner pairing space mV , i.e. mT = mV . (5.19) Proof. This follows from (5.16). To get relations between operators we assume (as in the binary case) uniqueness: for any T1, T2 : V → V it follows from v1|v2| . . . |T1vi| . . . vN−1|vN = v1|v2| . . . |T2vi| . . . vN−1|vN , (5.20) that T1 = T2 on any place i = 1, . . . , N. First, by analogy with the binary adjoint (5.8) we define N different adjoints for each operator T . Definition 5.4. Given a polyadic operator T : V → V on the polyadic inner pairing space HmK ,nK,mV ,kρ=1,N we define a polyadic adjoint as the set {T ⋆ij } of N operators T ⋆ij satisfying the following N equations T v1|v2|v3| . . . |vN = v1|T ⋆12 v2|v3| . . . |vN , v1|T v2|v3| . . . |vN = v1|v2|T ⋆23 v3| . . . |vN , ... v1|v2|v3| . . . T vN−1|vN = v1|v2|v3| . . . |T ⋆N−1,N vN , v1|v2|v3| . . . vN−1|T vN = T ⋆N,1 v1|v2|v3| . . . |vN , vi ∈ V. (5.21) In what follows, for the composition we will use the notation (T ⋆ij )⋆kl... ≡ T ⋆ij ⋆kl... . We have from (5.21) the N relations T ⋆12⋆23⋆34...⋆N−1,N ⋆N,1 = T, T ⋆23⋆34...⋆N−1,N ⋆N,1⋆12 = T, ... T ⋆N,1⋆12⋆23⋆34...⋆N−1,N = T, (5.22) which are called multistar cycles. Definition 5.5. We call the set of adjoint mappings (•⋆ij ) : T → T ⋆ij a polyadic involution, if they satisfy the multistar cycles (5.22).
24 STEVEN DUPLIJ If the inner pairing •| . . . |• has more than two places N ≥ 3, we have some additional structural issues, which do not exist in the binary case. First, we observe that the set of the adjointness relations (5.21) can be described in the framework of the associativity quiver approach introduced in DUPLIJ [2012] for polyadic representations. That is, for general N ≥ 3 in addition to (5.21) which corresponds to the so called Post-like associativity quiver (they will be called the Post-like adjointness relations), there also exist other sets. It is cum- bersome to write additional general formulas like (5.21) for other non-Post-like cases, while instead we give a clear example for N = 4. Example 5.6. The polyadic adjointness relations for N = 4 consist of the sets corresponding to different associativity quivers 1) Post-like adjointness relations T v1|v2|v3|v4 = v1|T ⋆12 v2|v3|v4 , v1|T v2|v3|v4 = v1|v2|T ⋆23 v3|v4 , v1|v2|T v3|v4 = v1|v2|v3|T ⋆34 v4 , v1|v2|v3|T v4 = T ⋆41 v1|v2|v3|v4 , 2) Non-Post-like adjointness relations T v1|v2|v3|v4 = v1|v2|v3|T ⋆14 v4 , v1|v2|v3|T v4 = v1|v2|T ⋆43 v3|v4 , v1|v2|T v3|v4 = v1|T ⋆32 v2|v3|v4 , v1|T v2|v3|v4 = T ⋆21 v1|v2|v3|v4 , (5.23) and the corresponding multistar cycles 1) Post-like multistar cycles T ⋆12⋆23⋆34⋆41 = T , T ⋆23⋆34⋆41⋆12 = T , T ⋆34⋆41⋆12⋆23 = T , T ⋆41⋆12⋆23⋆34 = T , 2) Non-Post-like multistar cycles T ⋆14⋆43⋆32⋆21 = T, T ⋆43⋆32⋆21⋆14 = T, T ⋆32⋆21⋆14⋆43 = T, T ⋆21⋆14⋆43⋆32 = T. (5.24) Thus, if the inner pairing has no symmetry, then both the Post-like and non-Post-like adjoints and corresponding multistar involutions are different. Second, in the case N ≥ 3 any symmetry of the multiplace inner pairing restricts the polyadic adjoint sets and multistar involutions considerably. Theorem 5.7. If the inner pairing with N ≥ 3 has the full permutation symmetry v1|v2| . . . |vN = σv1|σv2| . . . |σvN , ∀σ ∈ SN , (5.25) where SN is the symmetric group of N elements, then: 1) All the multistars coincide (⋆ij) = (⋆kl) := (∗) for any allowed i, j, k, l = 1, . . . , N; 2) All the operators are self-adjoint T = T ∗ . Proof. 1) In each adjointness relation from (5.21) we place the operator T on the l.h.s. to the first position and its multistar adjoint T ⋆ij to the second position, using the full permutation symmetry, which together with (5.20) gives the equality of all multistar operations. 2) We place the operator T on the l.h.s. to the first position and apply the derivation of the involution in the binary case to increasing cycles of size i ≤ N recursively, that is: For i = 2 T v1|v2|v3| . . . |vN = v1|T ∗ v2|v3| . . . |vN = T ∗ v2|v1|v3| . . . |vN = v2|T ∗∗ v1|v3| . . . |vN = T ∗∗ v1|v2|v3| . . . |vN , (5.26) then, using (5.20) we get T = T ∗∗ , (5.27) as in the standard binary case. However, for N ≥ 3 we have N higher cycles in addition.
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 25 For i = 3 T v1|v2|v3| . . . |vN = v1|T ∗ v2|v3| . . . |vN = T ∗ v2|v3|v1| . . . |vN = v2|T ∗∗ v3|v1| . . . |vN = T ∗∗ v3|v1|v2| . . . |vN = v3|T ∗∗∗ v1|v2| . . . |vN = T ∗∗∗ v1|v2|v3| . . . |vN , (5.28) which together with (5.20) gives T = T ∗∗∗ , (5.29) and after using (5.27) T = T ∗ . (5.30) Similarly, for an arbitrary length of the cycle i we obtain T = T i ∗ ∗ . . . ∗, which should be valid for each cycle recursively with i = 2, 3, . . . , N. Therefore, for any N ≥ 3 all the operators T are self-adjoint (5.30), while N = 2 is an exceptional case, when we have T = T ∗∗ (5.27) only. Now we show that imposing a partial symmetry on the polyadic inner pairing will give more inter- esting properties to the adjoint operators. Recall, that one of possible binary commutativity general- izations of (semi)groups to the polyadic case is the semicommutativity concept, when in the multipli- cation only the first and last elements are exchanged. Similarly, we introduce Definition 5.8. The polyadic inner pairing is called semicommutative, if v1|v2|v3| . . . |vN = vN |v2|v3| . . . |v1 , vi ∈ V. (5.31) Proposition 5.9. If the polyadic inner pairing is semicommutative, then for any operator T (satis- fying Post-like adjointness (5.21)) the last multistar operation (⋆N,1) is a binary involution and is a composition of all the previous multistars T ⋆N,1 = T ⋆12⋆23⋆34...⋆N−1,N , (5.32) T ⋆N,1⋆N,1 = T . (5.33) Proof. It follows from (5.21) and (5.31), that v1|v2|v3| . . . |T vN = T vN |v2|v3| . . . |v1 = vN |v2|v3| . . . |T ⋆12⋆23⋆34...⋆N−1,N v1 (5.34) = T ⋆12⋆23⋆34...⋆N−1,N v1|v2|v3| . . . |vN = T ⋆N,1 v1|v2|v3| . . . |vN , (5.35) which using (5.20) gives (5.32), (5.33) follows from the first multistar cocycle in (5.22). The adjointness relations (5.21) (of all kinds) together with (5.18) and (5.19) allows us to fix the arity shape of the polyadic operator algebra AT . We will assume that the arity of the operator multi- plication in AT coincides with the number of places of the inner pairing N (4.1) nT = N, (5.36) because it is in agreement with (5.21). Thus, the arity shape of the polyadic operator algebra becomes AT = K; {T } | σmK , κnK ; ηmT =mV , ωnT =N | θkF =kρ=1 , (5.37) Definition 5.10. We call the operator algebra AT which has the arity nT = N a nonderived polyadic operator algebra. Let us investigate some structural properties of AT and types of polyadic operators.
26 STEVEN DUPLIJ Remark 5.11. We can only define, but not derive, as in the binary case, the action of any multistar (⋆ij) on the product of operators, because in the nonderived nT -ary algebra we have a fixed number of operators in a product and sum, that is ℓ′ (nT − 1)+1 and ℓ′′ (mT − 1)+1, correspondingly, where ℓ′ is the number of nT -ary multiplications and ℓ′ is the number of mT -ary additions. Therefore, we cannot transfer (one at a time) all the polyadic operators from one place in the inner pairing to another place, as is done in the standard proof in the binary case. Taking this into account, as well as consistency under the multistar cycles (5.22), we arrive at Definition 5.12. The fixed multistar operation acts on the ℓ = 1 product of nT polyadic operators, depending on the sequential number of the multistar (⋆ij) (for the Post-like adjointness relations (5.21)) sij := i + j − 1 2 , if 3 ≤ i + j ≤ 2N − 1 N, if i j = N, sij = 1, 2, . . . , N − 1, N, (5.38) in the following way (ωnT [T1, T2, . . . , TnT −1, TnT ])⋆ij = ωnT T ⋆ij nT , T ⋆ij nT −1, . . . , T ⋆ij 2 , T ⋆ij 1 , if sij is odd, ωnT T ⋆ij 1 , T ⋆ij 2 , . . . , T ⋆ij nT −1, T ⋆ij nT , if sij is even. (5.39) A rule similar to (5.39) holds also for non-Post-like adjointness relations, but their concrete form depends of the corresponding non-Post-like associative quiver. Sometimes, to shorten notation, it will be more convenient to mark a multistar by the sequential number (5.38), such that (⋆ij) ⇒ ⋆sij , e.g. (⋆23) ⇒ (⋆2), (⋆N,1) ⇒ (⋆N ), etc. Also, in examples, for the ternary multiplication we will use the square brackets without the name of operation, if it is clear from the context, e.g. ω3 [T1, T2, T3] ⇒ [T1, T2, T3], etc. Example 5.13. In the lowest ternary case N = 3 we have 1) Post-like adjointness relations T v1|v2|v3 = v1|T ⋆1 v2|v3 , v1|T v2|v3 = v1|v2|T ⋆2 v3 , v1|v2|T v3 = T ⋆3 v1|v2|v3 , 2) Non-Post-like adjointness relations T v1|v2|v3 = v1|v2|T ⋆3 v3 , v1|v2|T v3 = v1|T ⋆2 v2|v3 , v1|T v2|v3 = T ⋆1 v1|v2|v3 , (5.40) and the corresponding multistar cycles 1) Post-like multistar cycles T ⋆1⋆2⋆3 = T , T ⋆2⋆3⋆1 = T , T ⋆3⋆1⋆2 = T , 2) Non-Post-like multistar cycles T ⋆3⋆2⋆1 = T, T ⋆2⋆1⋆3 = T, T ⋆1⋆3⋆2 = T. (5.41) Using (5.39) we obtain the following ternary conjugation rules ([T1, T2, T3])⋆1 = [T ⋆1 3 , T ⋆1 2 , T ⋆1 1 ] , (5.42) ([T1, T2, T3])⋆2 = [T ⋆2 1 , T ⋆2 2 , T ⋆2 3 ] , (5.43) ([T1, T2, T3])⋆3 = [T ⋆3 3 , T ⋆3 2 , T ⋆3 1 ] , (5.44) which are common for both Post-like and non-Post-like adjointness relations (5.40). Definition 5.14. A polyadic operator T is called self-adjoint, if all multistar operations are identities, i.e. (⋆ij) = id, ∀i, j.
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 27 5.2. Polyadic isometry and projection. Now we introduce polyadic analogs for the important types of operators: isometry, unitary, and (orthogonal) projection. Taking into account Remark 5.11, we again cannot move operators singly, and instead of proving the operator relations, as it is usually done in the binary case, we can only exploit some mnemonic rules to define the corresponding relations between polyadic operators. If the polyadic operator algebra AT contains a unit I and zero Z (see (5.5)–(5.6)) we define the conditions of polyadic isometry and orthogonality: Definition 5.15. A polyadic operator T is called a polyadic isometry, if it preserves the polyadic inner pairing T v1|T v2|T v3| . . . |T vN = v1|v2|v3| . . . |vN , (5.45) and satisfies ωnT [T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ] = I, + (N − 1) cycle permutations of multistars in the first (N − 1) terms. (5.46) Remark 5.16. If the multiplication in AT is derived and all multistars are equal, then the polyadic isometry operators satisfy some kind of N-regularity DUPLIJ AND MARCINEK [2002] or regular N-cocycle condition DUPLIJ AND MARCINEK [2001]. Proposition 5.17. The polyadic isometry operator T preserves the polyadic norm T v N = v N , ∀v ∈ V. (5.47) Proof. It follows from (4.9) and (5.45), that κnK   nK T v N , T v N , . . . , T v N   = κnK   nK v N , v N , . . . , v N   , (5.48) which gives (5.47), when nK = N. Definition 5.18. If for N polyadic operators Ti we have T1v1|T2v2|T3v3| . . . |TN vN = zK, ∀vi ∈ V, (5.49) where zK ∈ V is the zero of the underlying polyadic field KmK ,nK , then we say that Ti are (polyadi- cally) orthogonal, and they satisfy ωnT T ⋆N−1,N 1 , T ⋆N−2,N−1⋆N−1,N 2 , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N 3 , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N N−1 , TN = Z, (5.50) + (N − 1) cycle permutations of multistars in the first (N − 1) terms. (5.51) The polyadic analog of projection is given by Definition 5.19. If a polyadic operator P ∈ AT satisfies the polyadic idempotency condition ωnT   nT P , P , . . . P   = P, (5.52) then it is called a polyadic projection. By analogy with the binary case, polyadic projections can be constructed from polyadic isometry operators in a natural way.
28 STEVEN DUPLIJ Proposition 5.20. If T ∈ AT is a polyadic isometry, then P (1) T = ωnT [T , T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N ] , + (N − 1) cycle permutations of multistars in the last (N − 1) terms. (5.53) are the corresponding polyadic projections P (k) T , k = 1, . . . , N, satisfying (5.52). Definition 5.21. A polyadic operator T ∈ AT is called normal, if ωnT [T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ] = ωnT [T , T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N ] , + (N − 1) cycle permutations of multistars in the (N − 1) terms. (5.54) Proof. Insert (5.53) into (5.52) and use (5.46) together with nT -ary associativity. Definition 5.22. If all the polyadic projections (5.53) are equal to unity P (k) T = I, then the corre- sponding polyadic isometry operator T is called a polyadic unitary operator. It can be shown, that each polyadic unitary operator is querable (“polyadically invertible”), such that it has a querelement in AT . 5.3. Towards polyadic analog of C∗ -algebras. Let us, first, generalize the operator binary norm (5.1) to the polyadic case. This can be done, provided that a binary ordering on the underlying polyadic field KmK ,nK can be introduced. Definition 5.23. The polyadic operator norm • T : {T } → K is defined by T T = inf    M ∈ K | T v N ≤ µnK   nK −1 M, . . . , M, v N   , ∀v ∈ V    , (5.55) where • N is the polyadic norm in the inner pairing space HmK ,nK,mV ,kρ=1,N and µnK is the nK-ary multiplication in KmK ,nK . Definition 5.24. The polyadic operator norm is called submultiplicative, if ωnT [T1, T2, . . . , TnT ] T ≤ µnK [ T1 T , T2 T , . . . , TnK T ] , (5.56) nT = nK. (5.57) Definition 5.25. The polyadic operator norm is called subadditive, if ηmT [T1, T2, . . . , TnT ] T ≤ νmK [ T1 T , T2 T , . . . , TmK T ] , (5.58) mT = mK. (5.59) By analogy with the binary case, we have Definition 5.26. The polyadic operator algebra AT equipped with the submultiplicative norm • T a polyadic Banach algebra of operators BT . The connection between the polyadic norms of operators and their polyadic adjoints is given by Proposition 5.27. For polyadic operators in the inner pairing space HmK ,nK,mV ,kρ=1,N
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 29 1) The following N multi-C∗ -relations ωnT [T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ] N = µnK   nK T T , T T , . . . , T T   , + (N − 1) cycle permutations of (N − 1) terms with multistars, (5.60) take place, if nT = nK. 2) The polyadic norms of operator and its all adjoints coincide T ⋆i,j T = T T , ∀i, j ∈ 1, . . . , N. (5.61) Proof. Both statements follow from (5.21) and the definition of the polyadic operator norm (5.55). Therefore, we arrive to Definition 5.28. The operator Banach algebra BT satisfying the multi-C∗ -relations is called a polyadic operator multi-C∗ -algebra. The first example of a multi-C∗ -algebra (as in the binary case) can be constructed from one isometry operator (see Definition 5.15). Definition 5.29. A polyadic algebra generated by one isometry operator T satisfying (5.46) on the inner pairing space HmK ,nK ,mV ,kρ=1,N represents a polyadic Toeplitz algebra TmT ,nT and has the arity shape mT = mV , nT = N. Example 5.30. The ternary Toeplitz algebra T3,3 is represented by the operator T and relations [T ⋆1 , T ⋆3⋆1 , T ] = I, [T ⋆2 , T ⋆1⋆2 , T ] = I, [T ⋆3 , T ⋆2⋆3 , T ] = I. (5.62) Example 5.31. If the inner pairing is semicommutative (5.31), then (⋆3) can be eliminated by T ⋆3 = T ⋆1⋆2 , (5.63) T ⋆3⋆3 = T, (5.64) and the corresponding relations representing T3,3 become [T ⋆1 , T ⋆1 , T ] = I, [T ⋆2 , T ⋆1⋆2 , T ] = I, [T ⋆1⋆2 , T ⋆2 , T ] = I. (5.65) Let us consider M polyadic operators T1T2 . . . TM ∈ BT and the related partial (in the usual sense) isometries (5.52) which are mutually orthogonal (5.50). In the binary case, the algebra generated by M operators, such that the sum of the related orthogonal partial projections is unity, represents the Cuntz algebra OM CUNTZ [1977]. Definition 5.32. A polyadic algebra generated by M polyadic isometric operators T1T2 . . . TM ∈ BT satisfying η(ℓa) mT P (k) T1 , P (k) T2 . . . P (k) TM = I, k = 1, . . . , N, (5.66) where P (k) Ti are given by (5.53)and η (ℓa) mT is a “long polyadic addition” (5.16), represents a polyadic Cuntz algebra pOM|mT ,nT , which has the arity shape M = ℓa (mT − 1) + 1, (5.67)
30 STEVEN DUPLIJ where ℓa is number of “mT -ary additions”. Below we use the same notations, as in Example 5.13, also the ternary addition will be denoted by (+3) as follows: η3 [T1, T2, T3] ≡ T1 +3 T2 +3 T3. Example 5.33. In the ternary case mT = nT = 3 and one ternary addition ℓa = 1, we have M = 3 mutually orthogonal isometries T1, T2, T3 ∈ BT and N = 3 multistars (⋆i). In case of the Post-like multistar cocycles (5.41) they satisfy Isometry conditions [T ⋆1 i , T ⋆3⋆1 i , Ti] = I, [T ⋆2 i , T ⋆1⋆2 i , Ti] = I, [T ⋆3 i , T ⋆2⋆3 i , Ti] = I, i = 1, 2, 3, Orthogonality conditions T ⋆1 i , T ⋆3⋆1 j , Tk = Z, T ⋆2 i , T ⋆1⋆2 j , Tk = Z, T ⋆3 i , T ⋆2⋆3 j , Tk = Z, i, j, k = 1, 2, 3, i = j = k, (5.68) and the (sum of projections) relations [T1, T ⋆1 1 , T ⋆3⋆1 1 ] +3 [T2, T ⋆1 2 , T ⋆3⋆1 2 ] +3 [T3, T ⋆1 3 , T ⋆3⋆1 3 ] = I, (5.69) [T1, T ⋆2 1 , T ⋆1⋆2 1 ] +3 [T2, T ⋆2 2 , T ⋆1⋆2 2 ] +3 [T3, T ⋆2 3 , T ⋆1⋆2 3 ] = I, (5.70) [T1, T ⋆3 1 , T ⋆2⋆3 1 ] +3 [T2, T ⋆3 2 , T ⋆2⋆3 2 ] +3 [T3, T ⋆3 3 , T ⋆2⋆3 3 ] = I, (5.71) which represent the ternary Cuntz algebra pO3|3,3. Example 5.34. In the case where the inner pairing is semicommutative (5.31), we can eliminate the multistar (⋆3) by (5.63) and represent the two-multistar ternary analog of the Cuntz algebra pO3|3,3 by [T ⋆1 i , T ⋆2 i , Ti] = I, [T ⋆2 i , T ⋆1⋆2 i , Ti] = I, [T ⋆1⋆2 i , T ⋆2 i , Ti] = I, i = 1, 2, 3, T ⋆1 i , T ⋆2 j , Tk = Z, T ⋆1 i , T ⋆1⋆2 j , Tk = Z, T ⋆1⋆2 i , T ⋆2 j , Tk = Z, i, j, k = 1, 2, 3, i = j = k, (5.72) [T1, T ⋆1 1 , T ⋆1 1 ] +3 [T2, T ⋆1 2 , T ⋆1 2 ] +3 [T3, T ⋆1 3 , T ⋆1 3 ] = I, (5.73) [T1, T ⋆2 1 , T ⋆1⋆2 1 ] +3 [T2, T ⋆2 2 , T ⋆1⋆2 2 ] +3 [T3, T ⋆2 3 , T ⋆1⋆2 3 ] = I, (5.74) [T1, T ⋆1⋆2 1 , T ⋆2 1 ] +3 [T2, T ⋆1⋆2 2 , T ⋆2 2 ] +3 [T3, T ⋆1⋆2 3 , T ⋆2 3 ] = I. (5.75) 6. CONGRUENCE CLASSES AS POLYADIC RINGS Here we will show that the inner structure of the residue classes (congruence classes) over integers is naturally described by polyadic rings CELAKOSKI [1977], CROMBEZ [1972], LEESON AND BUTSON [1980], and then study some special properties of them. Denote a residue class (congruence class) of an integer a, modulo b by3 [[a]]b = {{a + bk} | k ∈ Z, a ∈ Z+, b ∈ N, 0 ≤ a ≤ b − 1} . (6.1) A representative element of the class [[a]]b will be denoted by xk = x (a,b) k = a + bk. Here we do not consider the addition and multiplication of the residue classes (congruence classes). Instead, we consider the fixed congruence class [[a]]b, and note that, for arbitrary a and b, it is not closed under binary operations. However, it can be closed with respect to polyadic operations. 3 We use for the residue class the notation [[a]]b , because the standard notations by one square bracket [a]b or ¯ab are busy by the n-ary operations and querelements, respectively.
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 31 6.1. Polyadic ring on integers. Let us introduce the m-ary addition and n-ary multiplication of representatives of the fixed congruence class [[a]]b by νm [xk1 , xk2 , . . . , xkm ] = xk1 + xk2 + . . . + xkm , (6.2) µn [xk1 , xk2 , . . . , xkn ] = xk1 xk2 . . . xkn , xki ∈ [[a]]b , ki ∈ Z, (6.3) where on the r.h.s. the operations are the ordinary binary addition and binary multiplication in Z. Remark 6.1. The polyadic operations (6.2)–(6.3) are not derived (see, e.g., GŁAZEK AND MICHALSKI [1984], MICHALSKI [1988]), because on the set {xki } one cannot define the binary semigroup struc- ture with respect to ordinary addition and multiplication. Derived polyadic rings which consist of the repeated binary sums and binary products were considered in LEESON AND BUTSON [1980]. Lemma 6.2. In case (m − 1) a b = I(m) (a, b) = I = integer (6.4) the algebraic structure [[a]]b | νm is a commutative m-ary group. Proof. The closure of the operation (6.2) can be written as xk1 + xk2 + . . . + xkm = xk0 , or ma + b (k1 + k2 + . . . + km) = a+bk0, and then k0 = (m − 1) a/b+(k1 + k2 + . . . + km), from (6.4). The (total) associativity and commutativity of νm follows from those of the addition in the binary Z. Each element xk has its unique querelement ˜x = x˜k determined by the equation (m − 1) xk + x˜k = xk, which (uniquely, for any k ∈ Z) gives ˜k = bk (2 − m) − (m − 1) a b . (6.5) Thus, each element is “querable” (polyadic invertible), and so [[a]]b | νm is a m-ary group. Example 6.3. For a = 2, b = 7 we have 8-ary group, and the querelement of xk is ˜x = x(−2−12k). Proposition 6.4. The m-ary commutative group [[a]]b | νm : 1) has an infinite number of neutral sequences for each element; 2) if a = 0, it has no “unit” (which is actually zero, because νm plays the role of “addition”); 3) in case of the zero congruence class [[0]]b the zero is xk = 0. Proof. 1) The (additive) neutral sequence ˜nm−1 of the length (m − 1) is defined by νm [˜nm−1, xk] = xk. Using (6.2), we take ˜nm−1 = xk1 + xk2 + . . . + xkm−1 = 0 and obtain the equation (m − 1) a + b (k1 + k2 + . . . + km−1) = 0. (6.6) Because of (6.4), we obtain k1 + k2 + . . . + km−1 = −I(m) (a, b) , (6.7) and so there is an infinite number of sums satisfying this condition. 2) The polyadic “unit”/zero z = xk0 = a + bk0 satisfies νm m−1 z, z, . . . , z, xk = xk for all xk ∈ [[a]]b (the neutral sequence ˜nm−1 consists of one element z only), which gives (m − 1) (a + bk0) = 0 having no solutions with a = 0, since a < b. 3) In the case a = 0, the only solution is z = xk=0 = 0. Example 6.5. In case a = 1, b = 2 we have m = 3 and I(3) (1, 2) = 1, and so from (6.6) we get k1 + k2 = −1, thus the infinite number of neutral sequences consists of 2 elements ˜n2 = xk + x−1−k, with arbitrary k ∈ Z.
32 STEVEN DUPLIJ Lemma 6.6. If an − a b = J(n) (a, b) = J = integer, (6.8) then [[a]]b | µn is a commutative n-ary semigroup. Proof. It follows from (6.3), that the closeness of the operation µn is xk1 xk2 . . . xkn = xk0 , which can be written as an + b (integer) = a + bk0 leading to (6.8). The (total) associativity and commutativity of µn follows from those of the multiplication in Z. Definition 6.7. A unique pair of integers (I, J) is called a (polyadic) shape invariants of the congru- ence class [[a]]b. Theorem 6.8. The algebraic structure of the fixed congruence class [[a]]b is a polyadic (m, n)-ring R[a,b] m,n = [[a]]b | νm, µn , (6.9) where the arities m and n are minimal positive integers (more or equal 2), for which the congruences ma ≡ a (mod b) , (6.10) an ≡ a (mod b) (6.11) take place simultaneously, fixating its polyadic shape invariants (I, J). Proof. By Lemma 6.2, 6.6 the set [[a]]b is a m-ary group with respect to “m-ary addition” νm and a n- ary semigroup with respect to “n-ary multiplication” µn, while the polyadic distributivity (1.1)–(1.3) follows from (6.2) and (6.3) and the binary distributivity in Z. Remark 6.9. For a fixed b ≥ 2 there are b congruence classes [[a]]b, 0 ≤ a ≤ b − 1, and therefore exactly b corresponding polyadic (m, n)-rings R [a,b] m,n, each of them is infinite-dimensional. Corollary 6.10. In case gcd (a, b) = 1 and b is prime, there exists the solution n = b. Proof. Follows from (6.11) and Fermat’s little theorem. Remark 6.11. We exclude from consideration the zero congruence class [[0]]b, because the arities of operations νm and µn cannot be fixed up by (6.10)–(6.11) becoming identities for any m and n. Since the arities are uncertain, their minimal values can be chosen m = n = 2, and therefore, it follows from (6.2) and (6.3), R [0,b] 2,2 = Z. Thus, in what follows we always imply that a = 0 (without using a special notation, e.g. R∗ , etc.). In TABLE 3 we present the allowed (by (6.10)–(6.11)) arities of the polyadic ring R [a,b] m,n and the corresponding polyadic shape invariants (I, J) for b ≤ 10. Let us investigate the properties of R [a,b] m,n in more detail. First, we consider equal arity polyadic rings and find the relation between the corresponding congruence classes. Proposition 6.12. The residue (congruence) classes [[a]]b and [[a′ ]]b′ which are described by the polyadic rings of the same arities R [a,b] m,n and R [a′,b′] m,n are related by b′ I′ a′ = bI a , (6.12) a′ + b′ J′ = (a + bJ)loga a′ . (6.13) Proof. Follows from (6.4) and (6.8). For instance, in TABLE 3 the congruence classes [[2]]5, [[3]]5, [[2]]10, and [[8]]10 are (6, 5)-rings. If, in addition, a = a′ , then the polyadic shapes satisfy I J = I′ J′ . (6.14)
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 33 6.2. Limiting cases. The limiting cases a ≡ ±1 (mod b) are described by Corollary 6.13. The polyadic ring of the fixed congruence class [[a]]b is: 1) multiplicative binary, if a = 1; 2) multiplicative ternary, if a = b − 1; 3) additive (b + 1)-ary in both cases. That is, the limiting cases contain the rings R [1,b] b+1,2 and R [b−1,b] b+1,3 , respectively. They correspond to the first row and the main diagonal of TABLE 3. Their intersection consists of the (3, 2)-ring R [1,2] 3,2 . Definition 6.14. The congruence classes [[1]]b and [[b − 1]]b are called the limiting classes, and the corresponding polyadic rings are named the limiting polyadic rings of a fixed congruence class. Proposition 6.15. In the limiting cases a = 1 and a = b − 1 the n-ary semigroup [[a]]b | µn : 1) has the neutral sequences of the form ¯nn−1 = xk1 xk2 . . . xkn−1 = 1, where xki = ±1; 2) has a) the unit e = xk=1 = 1, for the limiting class [[1]]b, b) the unit e− = xk=−1 = −1, if n is odd, for [[b − 1]]b, c) the class [[1]]2 contains both polyadic units e and e− ; 3) has the set of “querable” (polyadic invertible) elements which consist of ¯x = x¯k = ±1; 4) has in the “intersecting” case a = 1, b = 2 and n = 2 the binary subgroup Z2 = {1, −1}, while other elements have no inverses. Proof. 1) The (multiplicative) neutral sequence ¯nn−1 of length (n − 1) is defined by µn [¯nn−1, xk] = xk. It follows from (6.3) and cancellativity in Z, that ¯nn−1 = xk1 xk2 . . . xkn−1 = 1 which is (a + bk1) (a + bk2) . . . (a + bkn−1) = 1. (6.15) The solution of this equation in integers is the following: a) all multipliers are a + bki = 1, i = 1, . . . , n − 1; b) an even number of multipliers can be a + bki = −1, while the others are 1. 2) If the polyadic unit e = xk1 = a + bk1 exists, it should satisfy µm n−1 e, e, . . . , e, xk = xk ∀xk ∈ [[a]]b | µn , such that the neutral sequence ¯nn−1 consists of one element e only, and this leads to (a + bk1)n−1 = 1. For any n this equation has the solution a + bk1 = 1, which uniquely gives a = 1 and k1 = 0, thus e = xk1=0 = 1. If n is odd, then there exists a “negative unit” e− = xk1=−1 = −1, such that a + bk1 = −1, which can be uniquely solved by k1 = −1 and a = b − 1. The neutral sequence becomes ¯nn−1 = n−1 e− , e− , . . . , e− = 1, as a product of an even number of e− = −1. The intersection of limiting classes consists of one class [[1]]2, and therefore it contains both polyadic units e and e− . 3) An element xk in [[a]]b | µn is “querable”, if there exists its querelement ¯x = x¯k such that µn n−1 xk, xk, . . . , xk, ¯x = xk. Using (6.3) and the cancellativity in Z, we obtain the equa- tion (a + bk)n−2 a + b¯k = 1, which in integers has 2 solutions: a) (a + bk)n−2 = 1 and a + b¯k = 1, the last relation fixes up the class [[1]]b, and the arity of multiplication n = 2, and therefore the first relation is valid for all elements in the class, each of them has the same querelement ¯x = 1. This means that all elements in [[1]]b are “querable”, but only one element x = 1 has an inverse, which is also 1; b) (a + bk)n−2 = −1 and a + b¯k = −1. The second relation fixes the class [[b − 1]]b, and from the first relation we conclude that the arity n should be odd. In this case only one element −1 is “querable”, which has ¯x = −1, as a querelement. 4) The “intersecting” class [[1]]2 contains 2 “querable” elements ±1 which coincide with their inverses, which means that {+1, −1} is a binary subgroup (that is Z2) of the binary semigroup [[1]]2 | µ2 .
34 STEVEN DUPLIJ Corollary 6.16. In the non-limiting cases a = 1, b − 1, the n-ary semigroup [[a]]b | µn contains no “querable” (polyadic invertible) elements at all. Proof. It follows from (a + bk) = ±1 for any k ∈ Z or a = ±1 (mod b). TABLE 3. The polyadic ring R Z(a,b) m,n of the fixed residue class [[a]]b: arity shape. a b 2 3 4 5 6 7 8 9 10 1 m = 3 n = 2 I = 1 J = 0 m = 4 n = 2 I = 1 J = 0 m = 5 n = 2 I = 1 J = 0 m = 6 n = 2 I = 1 J = 0 m = 7 n = 2 I = 1 J = 0 m = 8 n = 2 I = 1 J = 0 m = 9 n = 2 I = 1 J = 0 m = 10 n = 2 I = 1 J = 0 m = 11 n = 2 I = 1 J = 0 2 m = 4 n = 3 I = 2 J = 2 m = 6 n = 5 I = 2 J = 6 m = 4 n = 3 I = 1 J = 1 m = 8 n = 4 I = 2 J = 2 m = 10 n = 7 I = 2 J = 14 m = 6 n = 5 I = 1 J = 3 3 m = 5 n = 3 I = 3 J = 6 m = 6 n = 5 I = 3 J = 48 m = 3 n = 2 I = 1 J = 1 m = 8 n = 7 I = 3 J = 312 m = 9 n = 3 I = 3 J = 3 m = 11 n = 5 I = 3 J = 24 4 m = 6 n = 3 I = 4 J = 12 m = 4 n = 2 I = 2 J = 2 m = 8 n = 4 I = 4 J = 36 m = 10 n = 4 I = 4 J = 28 m = 6 n = 3 I = 2 J = 6 5 m = 7 n = 3 I = 5 J = 20 m = 8 n = 7 I = 5 J = 11160 m = 9 n = 3 I = 5 J = 15 m = 10 n = 7 I = 5 J = 8680 m = 3 n = 2 I = 1 J = 2 6 m = 8 n = 3 I = 6 J = 30 m = 6 n = 2 I = 3 J = 3 7 m = 9 n = 3 I = 7 J = 42 m = 10 n = 4 I = 7 J = 266 m = 11 n = 5 I = 7 J = 1680 8 m = 10 n = 3 I = 8 J = 56 m = 6 n = 5 I = 4 J = 3276 9 m = 11 n = 3 I = 9 J = 72 Based on the above statements, consider in the properties of the polyadic rings R [a,b] m,n (a = 0) describing non-zero congruence classes (see Remark 6.11). Definition 6.17. The infinite set of representatives of the congruence (residue) class [[a]]b having fixed arities and form the (m, n)-ring R [a,b] m,n is called the set of (polyadic) (m, n)-integers (numbers) and denoted Z(m,n). Just obviously, for ordinary integers Z = Z(2,2), and they form the binary ring R [0,1] 2,2 . Proposition 6.18. The polyadic ring R [a,b] m,n is a (m, n)-integral domain. Proof. It follows from the definitions (6.2)–(6.3), the condition a = 0, and commutativity and can- cellativity in Z.
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 35 Lemma 6.19. There are no such congruence classes which can be described by polyadic (m, n)-field. Proof. Follows from Proposition 6.15 and Corollary 6.16. This statement for the limiting case [[1]]2 appeared in DUPLIJ AND WERNER [2015], while study- ing the ideal structure of the corresponding (3, 2)-ring. Proposition 6.20. In the limiting case a = 1 the polyadic ring R [1,b] b+1,2 can be embedded into a (b + 1, 2)-ary field. Proof. Because the polyadic ring R [1,b] b+1,2 of the congruence class [[1]]b is an (b + 1, 2)-integral domain by Proposition 6.18, we can construct in a standard way the correspondent (b + 1, 2)-quotient ring which is a (b + 1, 2)-ary field up to isomorphism, as was shown in CROMBEZ AND TIMM [1972]. By analogy, it can be called the field of polyadic rational numbers which have the form x = 1 + bk1 1 + bk2 , ki ∈ Z. (6.16) Indeed, they form a (b + 1, 2)-field, because each element has its inverse under multiplication (which is obvious) and additively “querable”, such that the equation for the querelement ¯x becomes νb+1 b x, x, . . . , x, ¯x = x which can be solved for any x, giving uniquely ¯x = − (b − 1) 1 + bk1 1 + bk2 . The introduced polyadic inner structure of the residue (congruence) classes allows us to extend various number theory problems by considering the polyadic (m, n)-integers Z(m,n) instead of Z. 7. EQUAL SUMS OF LIKE POWERS DIOPHANTINE EQUATION OVER POLYADIC INTEGERS First, recall the standard binary version of the equal sums of like powers Diophantine equation LANDER ET AL. [1967], EKL [1998]. Take the fixed non-negative integers p, q, l ∈ N0 , p ≤ q, and the positive integer unknowns ui, vj ∈ Z+, i = 1, . . . p + 1, j = 1, 1, . . . q + 1, then the Diophantine equation is p+1 i=1 ul+1 i = q+1 j=1 vl+1 j . (7.1) The trivial case, when ui = 0, vj = 0, for all i, j is not considered. We mark the solutions of (7.1) by the triple (l | p, q)r showing quantity of operations4 , where r (if it is used) is the order of the solution (ranked by the value of the sum) and the unknowns ui, vj are placed in ascending order ui ≤ ui+1, vj ≤ vj+1. Let us recall the Tarry-Escott problem (or multigrades problem) DORWART AND BROWN [1937]: to find the solutions to (7.1) for an equal number of summands on both sides of p = q and s equations simultaneously, such that l = 0, . . . , s. Known solutions exist for powers until s = 10, which are bounded such that s ≤ p (in our notations), see, also, NGUYEN [2016]. The solutions with highest powers s = p are the most interesting and called the ideal solutions BORWEIN [2002]. Theorem 7.1 (Frolov FROLOV [1889]). If the set of s Diophantine equations (7.1) with p = q for l = 0, . . . , s has a solution {ui, vi, i = 1, . . . p + 1}, then it has the solution {a + bui, a + bvi, i = 1, . . . p + 1}, where a, b ∈ Z are arbitrary and fixed. 4 In the binary case, the solutions of (7.1) are usually denoted by (l + 1 | p + 1, q + 1)r, which shows the number of summands on both sides and powers of elements LANDER ET AL. [1967]. But in the polyadic case (see below), the number of summands and powers do not coincide with l + 1, p + 1, q + 1, at all.
36 STEVEN DUPLIJ In the simplest case (1 | 0, 1), one term in l.h.s., one addition on the r.h.s. and one multiplication, the (coprime) positive numbers satisfying (7.1) are called a (primitive) Pythagorean triple. For the Fermat’s triple (l | 0, 1) with one addition on the r.h.s. and more than one multiplication l ≥ 2, there are no solutions of (7.1) , which is known as Fermat’s last theorem proved in WILES [1995]. There are many solutions known with more than one addition on both sides, where the highest number of multiplications till now is 31 (S. Chase, 2012). Before generalizing (7.1) for polyadic case we note the following. Remark 7.2. The notations in (7.1) are chosen in such a way that p and q are numbers of binary additions on both sides, while l is the number of binary multiplications in each term, which is natural for using polyadic powers DUPLIJ [2012]. 7.1. Polyadic analog of the Lander-Parkin-Selfridge conjecture. In LANDER ET AL. [1967], a generalization of Fermat’s last theorem was conjectured, that the solutions of (7.1) exist for small powers only, which can be formulated in terms of the numbers of operations as Conjecture 7.3 (Lander-Parkin-Selfridge LANDER ET AL. [1967]). There exist solutions of (7.1) in positive integers, if the number of multiplications is less than or equal than the total number of additions plus one 3 ≤ l ≤ lLSP = p + q + 1, (7.2) where p + q ≥ 2. Remark 7.4. If the equation (7.1) is considered over the binary ring of integers Z, such that ui, vj ∈ Z, it leads to a straightforward reformulation: for even powers it is obvious, but for odd powers all negative terms can be rearranged and placed on the other side. Let us consider the Diophantine equation (7.1) over polyadic integers Z(m,n) (i.e. over the polyadic (m, n)-ary ring RZ m,n) such that ui, vj ∈ RZ m,n. We use the “long products” µ (l) n and ν (l) m containing l operations, and also the “polyadic power” for an element x ∈ RZ m,n with respect to n-ary multiplica- tion DUPLIJ [2012] x l n = µ(l) n   l(n−1)+1 x, x, . . . , x   . (7.3) In the binary case, n = 2, the polyadic power coincides with (l + 1) power of an element x l 2 = xl+1 , which explains REMARK 7.2. In this notation the polyadic analog of the equal sums of like powers Diophantine equation has the form ν(p) m u l n 1 , u l n 2 , . . . , u l n p(m−1)+1 = ν(q) m v l n 1 , v l n 2 , . . . , v l n q(m−1)+1 , (7.4) where p and q are number of m-ary additions in l.h.s. and r.h.s. correspondingly. The solutions of (7.4) will be denoted by u1, u2, . . . , up(m−1)+1; v1, v2, . . . , vq(m−1)+1 . In the binary case m = 2, n = 2, (7.4) reduces to (7.1). Analogously, we mark the solutions of (7.4) by the polyadic triple (l | p, q)(m,n) r . Now the polyadic Pythagorean triple (1 | 0, 1)(m,n) , having one term on the l.h.s., one m-ary addition on the r.h.s. and one n-ary multiplication (elements are in the first polyadic power 1 n), becomes u 1 n 1 = νm v 1 n 1 , v 1 n 2 , . . . , v 1 n m . (7.5) Definition 7.5. The equation (7.5) solved by minimal u1, vi ∈ Z, i = 1, . . . , m can be named the polyadic Pythagorean theorem.
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 37 The polyadic Fermat’s triple (l | 0, 1)(m,n) has one term in l.h.s., one m-ary addition in r.h.s. and l (n-ary) multiplications u l n 1 = νm v l n 1 , v l n 2 , . . . , v l n m . (7.6) One may be interested in whether the polyadic analog of Fermat’s last theorem is valid, and if not, in which cases the analogy with the binary case can be sustained. Conjecture 7.6 (Polyadic analog of Fermat’s Last Theorem). The polyadic Fermat’s triple (7.6) has no solutions over the polyadic (m, n)-ary ring RZ m,n, if l ≥ 2, i.e. there are more than one n-ary multiplications. Its straightforward generalization leads to the polyadic version of the Lander-Parkin-Selfridge con- jecture, as Conjecture 7.7 (Polyadic Lander-Parkin-Selfridge conjecture). There exist solutions of the polyadic analog of the equal sums of like powers Diophantine equation (7.4) in integers, if the number of n-ary multiplications is less than or equal than the total number of m-ary additions plus one 3 ≤ l ≤ lpLP S = p + q + 1. (7.7) Below we will see a counterexample to both of the above conjectures. Example 7.8. Let us consider the (3, 2)-ring RZ 3,2 = Z | ν3, µ2 , where ν3 [x, y, z] = x + y + z + 2, (7.8) µ2 [x, y] = xy + x + y. (7.9) Note that this exotic polyadic ring is commutative and cancellative, having unit 0, no multiplicative inverses, and for any x ∈ RZ 3,2 its additive querelement ˜x = −x − 2, therefore Z | ν3 is a ternary group (as it should be). The polyadic power of any element is x l 2 = (x + 1)l+1 − 1. (7.10) 1) For RZ 3,2 the polyadic Pythagorean triple (1 | 0, 1)(3,2) in (7.5) now is u 1 2 = ν3 x 1 2 , y 1 2 , z 1 2 , (7.11) which, using (7.3), (7.9) and (7.10), becomes the (shifted) Pythagorean quadruple SPIRA [1962] (u + 1)2 = (x + 1)2 + (y + 1)2 + (z + 1)2 , (7.12) and it has infinite number of solutions, among which two minimal ones {u = 2; x = 0, y = z = 1} and {u = 14; x = 1, y = 9, z = 10} give 32 = 12 +22 +22 and 152 = 22 +102 +112 , correspondingly. 2) For this (3, 2)-ring RZ 3,2 the polyadic Fermat’s triple (l | 0, 1)(3,2) becomes (u + 1)l+1 = (x + 1)l+1 + (y + 1)l+1 + (z + 1)l+1 . (7.13) If the polyadic analog of Fermat’s last theorem 7.6 holds, then there are no solutions to (7.13) for more than one n-ary multiplication l ≥ 2. But this is the particular case, p = 0, q = 2, of the binary Lander-Parkin-Selfridge Conjecture 7.3 which now takes the form: the solutions to (7.13) exist, if l ≤ 3. Thus, as a counterexample to the polyadic analog of Fermat’s last theorem, we have two possible solutions with numbers of multiplications: l = 2, 3. In the case of l = 2 there exist two solutions: one well-known solution {u = 5; x = 2, y = 3, z = 4} giving 63 = 33 + 43 + 53 and another one giving 7093 = 1933 +4613 +6313 (J.-C. Meyrignac, 2000), while for l = 3 there exist an infinite number of solutions, and one of them (minimal) gives 4224814 = 958004 +2175194 +4145604 ELKIES [1988].
38 STEVEN DUPLIJ 3) The general polyadic triple (l | p, q)(3,2) , using (7.4), can be presented in the standard binary form (as (7.1)) 2p+1 i=1 (ui + 1)l+1 = 2q+1 j=1 (vi + 1)l+1 , ui, vj ∈ Z. (7.14) Let us apply the polyadic Lander-Parkin-Selfridge Conjecture 7.7 for this case: the solutions to (7.14) exist, if 3 ≤ l ≤ lpLSP = p + q + 1. But the binary Lander-Parkin-Selfridge Con- jecture 7.3, applied directly, gives 3 ≤ l ≤ lLSP = 2p + 2q + 1. So we should have coun- terexamples to the polyadic Lander-Parkin-Selfridge conjecture, when lpLSP < l ≤ lLSP . For in- stance, for p = q = 1, we have lpLSP = 3, while the (minimal) counterexample with l = 5 is {u1 = 3, u2 = 18, u3 = 21, v1 = 9, v2 = 14, v3 = 22} giving 36 + 196 + 226 = 106 + 156 + 236 SUBBA RAO [1934]. As it can be observed from Example 7.8, the arity shape of the polyadic ring RZ m,n is crucial in constructing polyadic analogs of the equal sums of like powers conjectures. We can make some general estimations assuming a special (more or less natural) form of its operations over integers. Definition 7.9. We call RZ m,n the standard polyadic ring, if the “leading terms” of its m-ary addition and n-ary multiplication are νm m x, x, . . . , x ∼ mx, (7.15) µn n x, x, . . . , x ∼ xn , x ∈ Z. (7.16) The polyadic ring RZ 3,2 from Example 7.8, as well as the congruence class polyadic ring R [a,b] m,n (6.9) are both standard. Using (7.3), we obtain approximate behavior of the polyadic power in the standard polyadic ring x l n ∼ xl(n−1)+1 , x ∈ Z, l ∈ N, n ≥ 2. (7.17) So increasing the arity of multiplication leads to higher powers, while increasing arity of addition gives more terms in sums. Thus, the estimation for the polyadic analog of the equal sums of like powers Diophantine equation (7.4) becomes (p (m − 1) + 1) xl(n−1)+1 ∼ (q (m − 1) + 1) xl(n−1)+1 , x ∈ Z. (7.18) Now we can apply the binary Lander-Parkin-Selfridge Conjecture 7.3 in the form: the solutions to (7.18) can exist if 3 ≤ l ≤ lLP S, where lLP S is an integer solution of (n − 1) lLP S = (p + q) (m − 1) + 1. (7.19) On the other hand the polyadic Lander-Parkin-Selfridge Conjecture 7.7 gives: the solutions to (7.18) can exist if 3 ≤ l ≤ lpLP S = p + q + 1. Note that (p + q) ≥ 2 now. An interesting question arises: which arities give the same limit, that is, when lpLP S = lLP S? Proposition 7.10. For any fixed number of additions in both sizes of the polyadic analog of the equal sums of like powers Diophantine equation (7.4) p + q ≥ 2, there exist limiting arities m0 and n0 (excluding the trivial binary case m0 = n0 = 2), for which the binary and polyadic Lander-Parkin- Selfridge conjectures coincide lpLP S = lLP S, such that m0 = 3 + p + q + (p + q + 1) k, (7.20) n0 = 2 + p + q + (p + q) k, k ∈ N0 . (7.21)
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 39 Proof. To equate lLP S = lpLP S = p + q + 1 we use (7.19) and solve in integers the equation (n0 − 1) (p + q + 1) = (p + q) (m0 − 1) + 1. (7.22) In the trivial case, m0 = n0 = 2, this is an identity, while the other solutions can be found from n0 (p + q + 1) = (p + q) m0 + 2, which gives (7.20)–(7.21). Corollary 7.11. In the limiting case lpLP S = lLP S the arity of multiplication always exceeds the arity of addition m0 − n0 = k + 1, k ∈ N0 , (7.23) and they start from m0 ≥ 5, n0 ≥ 4. The first allowed arities m0 and n0 are presented in TABLE 4. Their meaning is the following. Corollary 7.12. For the polyadic analog of the equal sums of like powers equation over the standard polyadic ring RZ m,n (with fixed p + q ≥ 2) the polyadic Lander-Parkin-Selfridge conjecture becomes weaker than the binary one lpLP S ≥ lLP S, if: 1) the arity of multiplication exceeds its limiting value n0 with fixed arity of the addition; 2) the arity of addition is lower, than its limiting value m0 with the fixed arity of multiplication. TABLE 4. The limiting arities m0 and n0, which give lpLP S = lLP S in (7.18). p + q = 2 p + q = 3 p + q = 4 m0 n0 m0 n0 m0 n0 5 4 6 5 7 6 8 6 10 8 12 10 11 8 14 11 17 14 14 10 18 14 22 18 Example 7.13. Consider the standard polyadic ring RZ m,n and fix the arity of addition m0 = 12, then take in (7.18) the total number of additions p + q = 4 (the last column in TABLE 4). We observe that the arity of multiplication n = 16, which exceeds the limiting arity n0 = 10 (corresponding to m0). Thus, we obtain lpLP S = 5 and lLP S = 3 by solving (7.19) in integers, and therefore the polyadic Lander-Parkin-Selfridge conjecture becomes now weaker than the binary one, and we do not obtain counterexamples to it, as in Example 7.8 (where the situation was opposite lpLP S = 3 and lLP S = 5, and they cannot be equal). A concrete example of the standard polyadic ring (Definition 7.9) is the polyadic ring of the fixed congruence class R [a,b] m,n considered in SECTION 6, because its operations (6.2)–(6.3) have the same straightforward behavior (7.15)–(7.16). Let us formulate the polyadic analog of the equal sums of like powers Diophantine equation (7.4) over R [a,b] m,n in terms of operations in Z. Using (6.2)–(6.3) and (7.17) for (7.4) we obtain p(m−1)+1 i=1 (a + bki)l(n−1)+1 = q(m−1)+1 j=1 (a + bkj)l(n−1)+1 , a, b, ki ∈ Z. (7.24) It is seen that the leading power behavior of both sides in (7.24) coincides with the general es- timation (7.18). But now the arity shape (m, n) is fixed by (6.10)–(6.11) and given in TABLE 3. Nevertheless, we can consider for (7.24) the polyadic analog of Fermat’s last theorem 7.6, the Lander- Parkin-Selfridge Conjecture 7.3 (solutions exist for l ≤ lLP S) and its polyadic version (Conjecture 7.7, solutions exist for l ≤ lpLP S), as in the estimations above. Let us consider some examples of solutions to (7.24).
40 STEVEN DUPLIJ Example 7.14. Let [[2]]3 be the congruence class, which is described by (4, 3)-ring R [2,3] 4,3 (see TABLE 3), and we consider the polyadic Fermat’s triple (l | 0, 5)(4,3) (7.6). Now the powers are lLP S = 8, lpLP S = 6, and for instance, if l = 2, we have solutions, because l < lpLP S < lLP S, and one of them is 145 = 4 · (−1)5 + 7 · 55 + 85 + 2 · 115 . (7.25) 7.2. Frolov’s Theorem and the Tarry-Escott problem. A special set of solutions to the polyadic Lander-Parkin-Selfridge Conjecture 7.7 can be generated, if we put p = q in (7.24), which we call equal-summand solutions5 , by exploiting the Tarry-Escott problem approach DORWART AND BROWN [1937] and Frolov’s Theorem 7.1. Theorem 7.15. If the set of integers ki ∈ Z solves the Tarry-Escott problem p(m−1)+1 i=1 kr i = p(m−1)+1 j=1 kr j , r = 1, . . . , s = l (n − 1) + 1, (7.26) then the polyadic equal sums of like powers equation with equal summands (7.4) has a solution over the polyadic (m, n)-ring R [a,b] m,n having the arity shape given by the following relations: 1) Inequality l (n − 1) + 1 ≤ p (m − 1) ; (7.27) 2) Equality p (m − 1) = 2l(n−1)+1 . (7.28) Proof. Using Frolov’s Theorem 7.1 applied to (7.26) we state that p(m−1)+1 i=1 (a + bki)r = p(m−1)+1 j=1 (a + bkj)r , r = 1, . . . , s = l (n − 1) + 1, (7.29) for any fixed integers a, b ∈ Z. This means, that (7.29), with ki (satisfying (7.26)) corresponds to a solution to the polyadic equal sums of like powers equation (7.4) for any congruence class [[a]]b. Nevertheless, the values a and b are fixed by the restrictions on the arity shape and the relations (6.4) and (6.8). 1) It is known that the Tarry-Escott problem can have a solution only when the powers are strongly less than the number of summands BORWEIN [2002], DORWART AND BROWN [1937], that is (l (n − 1) + 1) + 1 ≤ p (m − 1) + 1, which gives (7.27). 2) A special kind of solutions, when number of summands is equal to 2 into the number of powers, was found using the Thue–Morse sequence ALLOUCHE AND SHALLIT [1999], which always satisfies the bound (7.27), and in our notation it is (7.28). In both cases the relations (7.27) and (7.28) should be solved in positive integers and with m ≥ 2 and n ≥ 2, which can lead to non-unique solutions. Let us consider some examples which give solutions to the polyadic equal sums of like powers equation (7.4) with p = q over the polyadic (m, n)-ring R [a,b] m,n of the fixed congruence class [[a]]b. Example 7.16. 1) One of the first ideal (non-symmetric) solutions to the Tarry-Escott problem has 6 summands and 5 powers (A. Golden, 1944) 0r + 19r + 25r + 57r + 62r + 86r = 2r + 11r + 40r + 42r + 69r + 85r , r = 1, . . . , 5. (7.30) 5 The term “symmetric solution” is already taken and widely used BORWEIN [2002].
ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 41 We compare with (7.26) and obtain p (m − 1) = 5, (7.31) l (n − 1) = 4. (7.32) After ignoring binary arities we get m = 6, p = 1 and n = 3, l = 2. From Theorem 6.8 and TABLE 3 we observe the minimal choice a = 4 and b = 5. It follows from Frolov’s Theorem 7.1, that all equations in (7.30) have symmetry ki → a + bki = 4 + 5ki. Thus, we obtain the solution of the polyadic equal sums of like powers equation (7.4) for the fixed congruence class [[4]]5 in the form 45 + 995 + 1295 + 2895 + 3145 + 4345 = 145 + 595 + 2045 + 2145 + 3495 + 4295 . (7.33) It is seen from TABLE 3 that the arity shape (m = 6, n = 3) corresponds, e.g., to the congruence class [[4]]10 as well. Using Frolov’s theorem, we substitute in (7.30) ki → 4 + 10ki to obtain the solution in the congruence class [[4]]10 45 + 1945 + 2545 + 5745 + 6245 + 8645 = 245 + 1145 + 4045 + 4245 + 6945 + 8545 . (7.34) 2) To obtain the special kind of solutions to the Tarry-Escott problem we start with the known one with 8 summands and 3 powers (see, e.g., LEHMER [1947]) 0r +3r +5r +6r +9r +10r +12r +15r = 1r +2r +4r +7r +8r +11r +13r +14r , r = 1, 2, 3. (7.35) So we have the concrete solution to the system (7.26) with the condition (7.28) which now takes the form 8 = 23 , and therefore p (m − 1) = 7, (7.36) l (n − 1) = 2. (7.37) Excluding the trivial case containing binary arities, we have m = 8, p = 1 and n = 3, l = 1. It follows from Theorem 6.8 and TABLE 3, that a = 6 and b = 7, and so the polyadic ring is R [6,7] 8,3 . Using Frolov’s Theorem 7.1, we can substitute entries in (7.35) as ki → a + bki = 6 + 7ki in the equation with highest power r = 3 (which is relevant to our task) and obtain the solution of (7.4) for [[6]]7 as follows 63 +273 +413 +483 +693 +763 +903 +1113 = 133 +203 +343 +553 +623 +833 +973 +1043 . (7.38) We conclude that consideration of the Tarry-Escott problem and Frolov’s theorem over polyadic rings gives the possibility of obtaining many nontrivial solutions to the polyadic equal sums of like powers equation for fixed congruence classes. Acknowledgments. The author would like to express his deep gratitude and sincere thankfulness to Joachim Cuntz, Christopher Deninger, Grigorij Kurinnoj, Mike Hewitt, Jim Stasheff, Alexander Voronov, and Wend Werner for discussions, and to Dara Shayda for Mathematica programming help.
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ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 43 LEHMER, D. H. (1947). The Tarry-Escott problem. Scripta Math. 13, 37–41. LISTER, W. G. (1971). Ternary rings. Trans. Amer. Math. Soc. 154, 37–55. MICHALSKI, J. (1988). On J-derived polyadic groups. Mathematica (Cluj) 30 (53), 149–155. MIGNOT, F. (1976). Contrˆole dans les in´equations variationelles elliptiques. J. Funct. Anal. 22, 130–185. MONK, J. D. AND F. SIOSON (1966). m-semigroups, semigroups and function representations. Fundam. Math. 59, 233–241. NGUYEN, H. D. (2016). A new proof of the Prouhet-Tarry-Escott problem. Integers 16, A01, 1–9. POP, A. AND M. S. POP (2002). Some embeddings theorems for (n, 2)-rings. Bul. S¸tiint¸. Univ. Baia Mare, Ser. B, Fasc. Mat.-Inform. 18 (2), 311–316. RUSAKOV, S. A. (1998). Some Applications of n-ary Group Theory. Minsk: Belaruskaya navuka. SPIRA, R. (1962). The Diophantine equation x2 + y2 + z2 = m2. Am. Math. Mon. 69, 360–365. SUBBA RAO, K. (1934). On sums of sixth powers. J. London Math. Soc. s1-9 (3), 172–173. WILES, A. (1995). Modular elliptic curves and fermat’s last theorem. Ann. Math. 141 (3), 443–551. ZUPNIK, D. (1967). Polyadic semigroups. Publ. Math. (Debrecen) 14, 273–279. LIST OF TABLES 1 “Quantization” of arity ℓ-shapes 11 2 The arity signature and arity shape of polyadic algebraic structures. 21 3 The polyadic ring R Z(a,b) m,n of the fixed residue class [[a]]b: arity shape. 34 4 The limiting arities m0 and n0, which give lpLP S = lLP S in (7.18). 39 MATHEMATISCHES INSTITUTE, UNIVERSIT ¨AT M ¨UNSTER EINSTEINSTR. 62, D-48149 M ¨UNSTER, DEUTSCHLAND E-mail address: duplijs@math.uni-muenster.de, sduplij@gmail.com URL: http://homepages.spa.umn.edu/~duplij

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Sérgio Sacani
 

S.Duplij, Arity shape of polyadic algebraic structures (version 2)

  • 1. arXiv:1703.10132v2[math.RA]1May2017 ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES STEVEN DUPLIJ ABSTRACT. Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and lead to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, oper- ator norms, isometries and projections, as well as polyadic C∗ -algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with num- ber theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem are formulated. For the nonderived polyadic ring operations (polyadic numbers) neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov’s theorem for the Tarry-Escott problem is presented. CONTENTS INTRODUCTION 2 1. ONE SET POLYADIC “LINEAR” STRUCTURES 3 1.1. POLYADIC DISTRIBUTIVITY 3 1.2. POLYADIC RINGS AND FIELDS 4 2. TWO SET POLYADIC STRUCTURES 5 2.1. POLYADIC VECTOR SPACES 5 2.2. ONE-SET POLYADIC VECTOR SPACE 8 2.3. POLYADIC ALGEBRAS 9 3. MAPPINGS BETWEEN POLYADIC ALGEBRAIC STRUCTURES 12 3.1. POLYADIC FUNCTIONALS AND DUAL POLYADIC VECTOR SPACES 13 3.2. POLYADIC DIRECT SUM AND TENSOR PRODUCT 15 4. POLYADIC INNER PAIRING SPACES AND NORMS 19 APPLICATIONS 21 5. ELEMENTS OF POLYADIC OPERATOR THEORY 21 5.1. MULTISTARS AND POLYADIC ADJOINTS 23 5.2. POLYADIC ISOMETRY AND PROJECTION 27 5.3. TOWARDS POLYADIC ANALOG OF C∗-ALGEBRAS 28 6. CONGRUENCE CLASSES AS POLYADIC RINGS 30 6.1. POLYADIC RING ON INTEGERS 31 6.2. LIMITING CASES 33 7. EQUAL SUMS OF LIKE POWERS DIOPHANTINE EQUATION OVER POLYADIC INTEGERS 35 7.1. POLYADIC ANALOG OF THE LANDER-PARKIN-SELFRIDGE CONJECTURE 36 7.2. FROLOV’S THEOREM AND THE TARRY-ESCOTT PROBLEM 40 ACKNOWLEDGMENTS 41 REFERENCES 42 LIST OF TABLES 43 Date: March 29, 2017. 2010 Mathematics Subject Classification. 11D41, 11R04, 11R06, 17A42, 20N15, 47A05, 47L30, 47L70, 47L80.
  • 2. 2 STEVEN DUPLIJ INTRODUCTION The study of polyadic (higher arity) algebraic structures has a two-century long history, start- ing with works by Cayley, Sylvester, Kasner, Pr¨ufer, D¨ornte, Lehmer, Post, etc. They took a sin- gle set, closed under one (main) binary operation having special properties (the so called group- like structure), and “generalized” it by increasing the arity of that operation, which can then be called a polyadic operation and the corresponding algebraic structure polyadic as well1 . An “ab- stract way” to study polyadic algebraic structures is via the use of universal algebras defined as sets with different axioms (equational laws) for polyadic operations COHN [1965], GR ¨ATSER [1968], BERGMAN [2012]. However, in this language some important algebraic structures cannot be de- scribed, e.g. ordered groups, fields, etc. DENECKE AND WISMATH [2009]. Therefore, another “concrete approach” is to study examples of binary algebraic structures and then to “polyadize” them properly. This initiated the development of a corresponding theory of n-ary quasigroups BELOUSOV [1972], n-ary semigroups MONK AND SIOSON [1966], ZUPNIK [1967] and n-ary groups GAL’MAK [2003], RUSAKOV [1998] (for a more recent review, see, e.g., DUPLIJ [2012] and comprehensive list of references therein). The binary algebraic structures with two operations (addition and multiplication) on one set (the so-called ring-like structures) were later on generalized to (m, n)-rings CELAKOSKI [1977], CROMBEZ [1972], LEESON AND BUTSON [1980] and (m, n)- fields IANCU AND POP [1997], while these were investigated mostly in a more restrictive manner by considering particular cases: ternary rings (or (2, 3)-rings) LISTER [1971], (m, 2)-rings BOCCIONI [1965], POP AND POP [2002], as well as (3, 2)-fields DUPLIJ AND WERNER [2015]. In the case of one set, speaking informally, the “polyadization” of two operations’ “interaction” is straightforward, giving only polyadic distributivity which does not connect or restrict their arities. However, when the number of sets becomes greater than one, the “polyadization” turns out to be non- trivial, leading to special relations between the operation arities, and introduces additional (to arities) parameters, which allows us to classify them. We call the selection of such relations an arity shape and formulate the arity partial freedom principle that not all arities of operations during “polyadization” of binary operations are possible. In this paper we consider two-set algebraic structures in the “concrete way” and provide the con- sequent “polyadization” of binary operations on them for the so-called module-like structures (vector spaces) and algebra-like structures (algebras and inner product spaces). The “polyadization” of the binary scalar multiplication is made in terms of multiactions introduced in DUPLIJ [2012], having special arity shapes parametrized by the number of intact elements (ℓid) in the corresponding multi- actions. Then we “polyadize” the related constructions, as dual vector spaces and direct sums, and also tensor products, and show that, as opposed to the binary case, they can be implemented in spaces of different arity signatures. The “polyadization” of inner product spaces and related norms gives additional arity shapes and restrictions. In the resulting TABLE 2 we present the arity signatures and shapes of the polyadic algebraic structures under consideration. In the application part we note some starting points for polyadic operator theory by introducing multistars and polyadic analogs of adjoints, operator norms, isometries and projections. It is proved (Theorem 5.7) that, if the polyadic inner pairing (the analog of the inner product) is symmetric, then all multistars coincide and all polyadic operators are self-adjoint (as opposed to the binary case). The polyadic analogs of C∗ -algebras, Toeplitz algebras and Cuntz algebras are presented in terms of the polyadic operators introduced here, and the ternary example is given. Another application is connected with number theory: we show that the internal structure of the congruence classes is described by a polyadic ring having a special arity signature (TABLE 3), and we call them polyadic integers (numbers) Z(m,n) (Definition 6.17). They are classified by polyadic shape invariants, and the relations between them giving the same arity signature are established. Also the 1 We use the term “polyadic” in this sense only, while there are other uses in the literature (see, e.g., HALMOS [1962]).
  • 3. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 3 limiting cases are analyzed, and it is shown that in one such a case the polyadic rings can be embedded into polyadic fields with binary multiplication, which leads to the so-called polyadic rational num- bers CROMBEZ AND TIMM [1972]. Then we consider the Diophantine equations over these polyadic rings in a straightforward manner: we change only arities of operations (“additions” and “multipli- cations”), but save their mutual “interaction”. In this way we try to “polyadize” the equal sums of like powers equation and formulate the polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem LANDER ET AL. [1967]. It is shown, that in the simplest case, when the polyadic “addition” and “multiplication” are nonderived (e.g., polyadic numbers), neither conjecture is valid, and counterexamples are presented. Finally, we apply Frolov’s theorem for the Tarry-Escott problem DORWART AND BROWN [1937], NGUYEN [2016] over polyadic rings to obtain new solu- tions to the equal sums of like powers equation for a fixed congruence classes. 1. ONE SET POLYADIC “LINEAR” STRUCTURES We use concise notations from our previous work on polyadic structures DUPLIJ [2012, 2016]. Take a non-empty set A, then n-tuple (or polyad) consisting of the elements (a1, . . . , an), ai ∈ A, is denoted by bold letter (a) taking it values in the Cartesian product A×n . If the number of elements in the n-tuple is important, we denote it manifestly a(n) , and an n-tuple with equal elements is denoted by (an ). On the Cartesian product A×n one can define a polyadic operation µn : A×n → A, and use the notation µn [a]. A polyadic structure A is a set A which is closed under polyadic operations, and a polyadic signature is the selection of their arities. For formal definitions, see, e.g., COHN [1965]. 1.1. Polyadic distributivity. Let us consider a polyadic structure with two operations on the same set A: the “chief” (multiplication) n-ary operation µn : An → A and the additional m-ary operation νm : Am → A, that is A | µn, νm . If there are no relations between µn and νm, then nothing new, as compared with the polyadic structures having a single operation A | µn or A | νm , can be said. Informally, the “interaction” between operations can be described using the important relation of distributivity (an analog of a · (b + c) = a · b + a · c, a, b, c ∈ A in the binary case). Definition 1.1. The polyadic distributivity for the operations µn and νm (no additional properties are implied for now) consists of n relations µn [νm [a1, . . . am] , b2, b3, . . . bn] = νm [µn [a1, b2, b3, . . . bn] , µn [a2, b2, b3, . . . bn] , . . . µn [am, b2, b3, . . . bn]] (1.1) µn [b1, νm [a1, . . . am] , b3, . . . bn] = νm [µn [b1, a1, b3, . . . bn] , µn [b1, a2, b3, . . . bn] , . . . µn [b1, am, b3, . . . bn]] (1.2) ... µn [b1, b2, . . . bn−1, νm [a1, . . . am]] = νm [µn [b1, b2, . . . bn−1, a1] , µn [b1, b2, . . . bn−1, a2] , . . . µn [b1, b2, . . . bn−1, am]] , (1.3) where ai, bj ∈ A. It is seen that the operations µn and νm enter into (1.1)-(1.3) in a non-symmetric way, which allows us to distinguish them: one of them (µn, the n-ary multiplication) “distributes” over the other one νm, and therefore νm is called the addition. If only some of the relations (1.1)-(1.3) hold, then such distributivity is partial (the analog of left and right distributivity in the binary case). Obviously, the operations µn and νm need have nothing to do with ordinary multiplication (in the binary case denoted by µ2 =⇒ (·)) and addition (in the binary case denoted by ν2 =⇒ (+)), as in the below example.
  • 4. 4 STEVEN DUPLIJ Example 1.2. Let A = R, n = 2, m = 3, and µ2 [b1, b2] = bb2 1 , ν3 [a1, a2, a3] = a1a2a3 (product in R). The partial distributivity now is (a1a2a3)b2 = ab2 1 ab2 2 ab2 3 (only the first relation (1.1) holds). 1.2. Polyadic rings and fields. Here we briefly remind the reader of one-set (ring-like) polyadic structures (informally). Let both operations µn and νm be (totally) associative, which (in our defi- nition DUPLIJ [2012]) means independence of the composition of two operations under placement of the internal operations (there are n and m such placements and therefore (n + m) corresponding relations) µn [a, µn [b] , c] = invariant, (1.4) νm [d, νm [e] , f] = invariant, (1.5) where the polyads a, b, c, d, e, f have corresponding length, and then both A | µn | assoc and A | νm | assoc are polyadic semigroups Sn and Sm. A commutative semigroup A | νm | assoc, comm is defined by νm [a] = νm [σ ◦ a], for all σ ∈ Sn, where Sn is the symmetry group. If the equation νm [a, x, b] = c is solvable for any place of x, then A | νm | assoc, solv is a polyadic group Gm, and such x = ˜c is called a (additive) querelement for c, which defines the (additive) unary queroperation ˜ν1 by ˜ν1 [c] = ˜c. Definition 1.3. A polyadic (m, n)-ring Rm,n is a set A with two operations µn : An → A and νm : Am → A, such that: 1) they are distributive (1.1)-(1.3); 2) A | µn | assoc is a polyadic semigroup; 3) A | νm | assoc, comm, solv is a commutative polyadic group. It is obvious that a (2, 2)-ring R2,2 is an ordinary (binary) ring. Polyadic rings have much richer structure and can have unusual properties CELAKOSKI [1977], CROMBEZ [1972], ˇCUPONA [1965], LEESON AND BUTSON [1980]. If the multiplicative semigroup A | µn | assoc is commutative, µn [a] = µn [σ ◦ a], for all σ ∈ Sn, then Rm,n is called a commutative polyadic ring, and if it contains the identity, then Rm,n is a (polyadic) (m, n)-semiring. If the distributivity is only partial, then Rm,n is called a polyadic near-ring. Introduce in Rm,n additive and multiplicative idempotent elements by νm [am ] = a and µn [bn ] = b, respectively. A zero z of Rm,n is defined by µn [z, a] = z for any a ∈ An−1 , where z can be on any place. Evidently, a zero (if exists) is a multiplicative idempotent and is unique, and, if a polyadic ring has an additive idempotent, it is a zero LEESON AND BUTSON [1980]. Due to the distributivity (1.1)-(1.3), there can be at most one zero in a polyadic ring. If a zero z exists, denote A∗ = A {z}, and observe that (in distinction to binary rings) A∗ | µn | assoc is not a polyadic group, in general. In the case where A∗ | µn | assoc is a commutative n-ary group, such a polyadic ring is called a (polyadic) (m, n)-field and Km,n (“polyadic scalars”) (see LEESON AND BUTSON [1980], IANCU AND POP [1997]). A multiplicative identity e in Rm,n is a distinguished element e such that µn a, en−1 = a, (1.6) for any a ∈ A and where a can be on any place. In binary rings the identity is the only neutral element, while in polyadic rings there can exist many neutral (n − 1)-polyads e satisfying µn [a, e] = a, (1.7) for any a ∈ A which can also be on any place. The neutral polyads e are not determined uniquely. Obviously, the polyad (en−1 ) is neutral. There exist exotic polyadic rings which have no zero, no identity, and no additive idempotents at all (see, e.g., CROMBEZ [1972]), but, if m = 2, then a zero always exists LEESON AND BUTSON [1980].
  • 5. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 5 Example 1.4. Let us consider a polyadic ring R3,4 generated by 2 elements a, b and the relations µ4 a4 = a, µ4 a3 , b = b, µ4 a2 , b2 = a, µ4 a, b3 = b, µ4 b4 = a, (1.8) ν3 a3 = b, ν3 a2 , b = a, ν3 a, b2 = b, ν3 b3 = a, (1.9) which has a multiplicative idempotent a only, but has no zero and no identity. Proposition 1.5. In the case of polyadic structures with two operations on one set there are no con- ditions between arities of operations which could follow from distributivity (1.1)-(1.3) or the other relations above, and therefore they have no arity shape. Such conditions will appear below, when we consider more complicated universal algebraic struc- tures with two or more sets with operations and relations. 2. TWO SET POLYADIC STRUCTURES 2.1. Polyadic vector spaces. Let us consider a polyadic field KmK ,nK = K | σmK , κnK (“polyadic scalars”), having mK-ary addition σmK : KmK → K and nK-ary multiplication κnK : KnK → K, and the identity eK ∈ K, a neutral element with respect to multiplication κnK enK −1 K , λ = λ, for all λ ∈ K. In polyadic structures, one can introduce a neutral (nK − 1)-polyad (identity polyad for “scalars”) eK ∈ KnK−1 by κnK [eK, λ] = λ. (2.1) where λ ∈ K can be on any place. Next, take a mV -ary commutative (abelian) group V | νmV , which can be treated as “polyadic vectors” with mV -ary addition νmV : VmV → V. Define in V | νmV an additive neutral element (zero) zV ∈ V by νmV zmV −1 V , v = v (2.2) for any v ∈ V, and a “negative vector” ¯v ∈ V as its querelement νmV [aV , ¯v, bV ] = v, (2.3) where ¯v can be on any place in the l.h.s., and aV , bV are polyads in V. Here, instead of one neutral element we can also introduce the (mV − 1)-polyad zV (which may not be unique), and so, for a zero polyad (for “vectors”) we have νmV [zV , v] = v, ∀v ∈ V, (2.4) where v ∈ V can be on any place. The “interaction” between “polyadic scalars” and “polyadic vectors” (the analog of binary multiplication by a scalar λv) can be defined as a multiaction (kρ-place action) introduced in DUPLIJ [2012] ρkρ : Kkρ × V −→ V. (2.5) The set of all multiactions form a nρ-ary semigroup Sρ under composition. We can “normalize” the multiactions in a similar way, as multiplace representations DUPLIJ [2012], by (an analog of 1v = v, v ∈ V, 1 ∈ K) ρkρ    eK ... eK v    = v, (2.6) for all v ∈ V, where eK is the identity of KmK ,nK . In the case of an (ordinary) 1-place (left) action (as an external binary operation) ρ1 : K × V → V, its consistency with the polyadic field multiplication κnK under composition of the binary operations ρ1 {λ|a} gives a product of the same arity nρ = nK, (2.7)
  • 6. 6 STEVEN DUPLIJ that is (a polyadic analog of λ (µv) = (λµ) v, v ∈ V, λ, µ ∈ K) ρ1 {λ1|ρ1 {λ2| . . . |ρ1 {λnK |v}} . . .} = ρ1 {κnK [λ1, λ2, . . . λnK ] |v} , λ1, . . . , λn ∈ K, v ∈ V. (2.8) In the general case of kρ-place actions, the multiplication in the nρ-ary semigroup Sρ can be defined by the changing arity formula DUPLIJ [2012] (schematically) ρkρ nρ    λ1 ... λkρ . . . ρkρ    λkρ(nρ−1) ... λkρnρ v    . . .    = ρkρ    κnK [λ1, . . . λnK ] , ... κnK λnK(ℓµ−1), . . . λnKℓµ    ℓµ λnKℓµ+1, ... λnKℓµ+ℓid    ℓid v    , (2.9) where ℓµ and ℓid are both integers. The associativity of (2.9) in each concrete case can be achieved by applying the associativity quiver concept from DUPLIJ [2012]. Definition 2.1. The ℓ-shape is a pair (ℓµ, ℓid) , where ℓµ is the number of multiplications and ℓid is the number of intact elements in the composition of operations. It follows from (2.9), Proposition 2.2. The arities of the polyadic field KmK ,nK , the arity nρ of the multiaction semigroup Sρ and the ℓ-shape of the composition satisfy kρnρ = nKℓµ + ℓid, (2.10) kρ = ℓµ + ℓid. (2.11) We can exclude ℓµ or ℓid and obtain nρ = nK − nK − 1 kρ ℓid, nρ = nK − 1 kρ ℓµ + 1, (2.12) respectively, where nK−1 kρ ℓid ≥ 1 and nK−1 kρ ℓµ ≥ 1 are integers. The following inequalities hold 1 ≤ ℓµ ≤ kρ, 0 ≤ ℓid ≤ kρ − 1, ℓµ ≤ kρ ≤ (nK − 1) ℓµ, 2 ≤ nρ ≤ nK. (2.13) Remark 2.3. The formulas (2.12) coincide with the arity changing formulas for heteromorphisms DUPLIJ [2012] applied to (2.9). It follows from (2.10), that the ℓ-shape is determined by the arities and number of places kρ by ℓµ = kρ (nρ − 1) nK − 1 , ℓid = kρ (nK − nρ) nK − 1 . (2.14) Because we have two polyadic “additions” νmV and σmK , we need to consider, how the multiaction ρkρ “distributes” between each of them. First, consider distributivity of the multiaction ρkρ with respect to “vector addition” νmV (a polyadic analog of the binary λ (v + u) = λv + λu, v, u ∈ V, λ, µ ∈ K) ρkρ    λ1 ... λkρ νmV [v1, . . . , vmV ]    = νmV  ρkρ    λ1 ... λkρ v1    , . . . , ρkρ    λ1 ... λkρ vmV      . (2.15) Observe that here, in distinction to (2.9), there is no connection between the arities mV and kρ.
  • 7. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 7 Secondly, the distributivity of the multiaction ρkρ (“multiplication by scalar”) with respect to the “field addition” (a polyadic analog of (λ + µ) v = λv + µv, v ∈ A, λ, µ ∈ K) has a form similar to (2.9) (which can be obtained from the changing arity formula DUPLIJ [2012]) ρkρ nρ    λ1 ... λkρ . . . ρkρ    λkρ(nρ−1) ... λkρnρ v    . . .    = ρkρ    σmK [λ1, . . . λmK ] , ... σmK λmK (ℓ′ µ−1), . . . λmK ℓ′ µ    ℓ′ µ λmK ℓ′ µ+1, ... λmK ℓ′ µ+ℓid    ℓ′ id v    , (2.16) where ℓ′ ρ and ℓ′ id are the numbers of multiplications and intact elements in the resulting multiaction, respectively. Here the arities are not independent as in (2.15), and so we have Proposition 2.4. The arities of the polyadic field KmK ,nK , the arity nρ of the multiaction semigroup Sρ and the ℓ-shape of the distributivity satisfy kρnρ = mKℓ′ µ + ℓ′ id, (2.17) kρ = ℓ′ µ + ℓ′ id. (2.18) It follows from (2.17)–(2.18) nρ = mK − mK − 1 kρ ℓ′ id, nρ = mK − 1 kρ ℓ′ µ + 1. (2.19) Here mK −1 kρ ℓ′ id ≥ 1 and mK −1 kρ ℓ′ µ ≥ 1 are integers, and we have the inequalities 1 ≤ ℓ′ µ ≤ kρ, 0 ≤ ℓ′ id ≤ kρ − 1, ℓ′ µ ≤ kρ ≤ (mK − 1) ℓ′ µ, 2 ≤ nρ ≤ mK. (2.20) Now, the ℓ-shape of the distributivity is fully determined from the arities and number of places kρ by the arity shape formulas ℓ′ ρ = kρ (nρ − 1) mK − 1 , ℓ′ id = kρ (mK − nρ) mK − 1 . (2.21) It follows from (2.20) that: Corollary 2.5. The arity nρ of the multiaction semigroup Sρ is less than or equal to the arity of the field addition mK. Definition 2.6. A polyadic (K)-vector (“linear”) space over a polyadic field is the 2-set 4-operation algebraic structure VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ , (2.22) such that the following axioms hold: 1) K | σmK , κnK is a polyadic (mK, nK)-field KmK ,nK ; 2) A | νmV is a commutative mV -ary group; 3) ρkρ | composition is a nρ-ary semigroup Sρ; 4) Distributivity of the multiaction ρkρ with respect to the “vector addition” νmV (2.15); 5) Distributivity of ρkρ with respect to the “scalar addition” σmK (2.16); 6) Compatibility of ρkρ with the “scalar multiplication” κnK (2.9); 7) Normalization of the multiaction ρkρ (2.6).
  • 8. 8 STEVEN DUPLIJ All of the arities in (2.22) are independent and can be chosen arbitrarily, but they fix the ℓ-shape of the multiaction composition (2.9) and the distributivity (2.16) by (2.14) and (2.21), respectively. Note that the main distinction from the binary case is the possibility for the arity nρ of the multiaction semigroup Sρ to be arbitrary. Definition 2.7. A polyadic K-vector subspace is Vsub mK ,nK,mV ,kρ = K; Vsub | σmK , κnK ; νmV | ρkρ , (2.23) where the subset Vsub ⊂ V is closed under all operations σmK , κnK , νmV , ρkρ and the axioms 1)-7). Let us consider a subset S = {v1, . . . , vdV } ⊆ V (of dV “vectors”), then a polyadic span of S is (a “linear combination”) Spanλ pol (v1, . . . , vdV ) = {w} , (2.24) w = νℓν mV  ρkρ    λ1 ... λkρ v1    , . . . , ρkρ    λ(dV −1)kρ ... λdV kρ vs      , (2.25) where (dV · kρ) “scalars” play the role of coefficients (or coordinates as columns consisting of kρ elements from the polyadic field KmK ,nK ), and the number of “vectors” s is connected with the “number of mV -ary additions” ℓν by dV = ℓν (mV − 1) + 1, (2.26) while Spanλ pol S is the set of all “vectors” of this form (2.24) (we consider here finite “sums” only). Definition 2.8. A polyadic span S = {v1, . . . , vdV } ⊆ V is nontrivial, if at least one multiaction ρkρ in (2.24) is nonzero. Since polyadic fields and groups may not contain zeroes, we need to redefine the basic notions of equivalences. Let us take two different spans of the same set S. Definition 2.9. A set {v1, . . . , vdV } is called “linear” polyadic independent, if from the equality of nontrivial spans, as Spanλ pol (v1, . . . , vdV ) = Spanλ′ pol (v1, . . . , vdV ), it follows that all λi = λ′ i, i = 1, . . . , dV kρ. Definition 2.10. A set {v1, . . . , vdV } is called a polyadic basis of a polyadic vector space VmK nK mV kρ , if it spans the whole space Spanλ pol (v1, . . . , vdV ) = V. In other words, any element of V can be uniquely presented in the form of the polyadic “linear combination” (2.24). If a polyadic vector space VmK nK mV kρ has a finite basis {v1, . . . , vdV }, then any another basis v′ 1, . . . , v′ dV has the same number of elements. Definition 2.11. The number of elements in the polyadic basis {v1, . . . , vdV } is called the polyadic dimension of VmK ,nK ,mV ,kρ. Remark 2.12. The so-called 3-vector space introduced and studied in DUPLIJ AND WERNER [2015], corresponds to VmK =3,nK=2,mV =3,kρ=1. 2.2. One-set polyadic vector space. A particular polyadic vector space is important: consider V = K, νmV = σmK and mV = mK, which gives the following one-set “linear” algebraic structure (we call it a one-set polyadic vector space) KmK ,nK,kρ = K | σmK , κnK | ρλ kρ , (2.27)
  • 9. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 9 where now the multiaction ρλ kρ    λ1 ... λkρ λ    , λ, λi ∈ K, (2.28) acts on K itself (in some special way), and therefore can be called a regular multiaction. In the binary case nK = mK = 2, the only possibility for the regular action is the multiplication (by “scalars”) in the field ρλ 1 {λ1| λ} = κ2 [λ1λ] (≡ λ1λ), which obviously satisfies the axioms 4)-7) of a vector space in Definition 2.6. In this way we arrive at the definition of the binary field K ≡ K2,2 = K | σ2, κ2 , and so a one-set binary vector space coincides with the underlying field KmK =2,nK=2,kρ=1 = K, or as it is said “a field is a (one-dimensional) vector space over itself”. Remark 2.13. In the polyadic case, the regular multiaction ρλ kρ can be chosen, as any (additional to σmK , κnK ) function satisfying axioms 4)-7) of a polyadic vector space and the number of places kρ and the arity of the semigroup of multiactions Sρ can be arbitrary, in general. Also, ρλ kρ can be taken as a some nontrivial combination of σmK , κnK satisfying axioms 4)-7) (which admits a nontrivial “multiplication by scalars”). In the simplest regular (similar to the binary) case, ρλ kρ    λ1 ... λkρ λ    = κℓκ nK λ1, . . . , λkρ, λ , (2.29) where ℓκ is the number of multiplications κnK , and the number of places kρ is now fixed by kρ = ℓκ (nK − 1) , (2.30) while λ in (2.29) can be on any place due the commutativity of the field multiplication κnK . Remark 2.14. In general, the one-set polyadic vector space need not coincide with the underlying polyadic field, KmK ,nK ,kρ = KnK mK (as opposed to the binary case), but can have a more complicated structure which is determined by an additional multiplace function, the multiaction ρλ kρ . 2.3. Polyadic algebras. By analogy with the binary case, introducing an additional operation on vectors, a multiplication which is distributive and “linear” with respect to “scalars”, leads to a polyadic generalization of the (associative) algebra notion CARLSSON [1980]. Here, we denote the second (except for the ’scalars’ K) set by A (instead of V above), on which we define two operations: mA-ary “addition” νmA : A×mA → A and nA-ary “multiplication” µnA : A×nA → A. To interpret the nA-ary operation as a true multiplication, the operations µnA and νmA should satisfy polyadic distributivity (1.1)–(1.3) (an analog of (a + b) · c = a · c + b · c, with a, b, c ∈ A). Then we should consider the “interaction” of this new operation µnA with the multiaction ρkρ (an analog of the “compatibility with scalars” (λa) · (µb) = (λµ) a · b, a, b ∈ A, λ, µ ∈ K). In the most general case, when all arities are
  • 10. 10 STEVEN DUPLIJ arbitrary, we have the polyadic compatibility of µnA with the field multiplication κnK as follows µnA  ρkρ    λ1 ... λkρ a1    , . . . , ρkρ    λkρ(nA−1) ... λkρnA anA      = ρkρ    κnK [λ1, . . . , λnK ] , ... κnK λnK (ℓ′′ µ−1), . . . , λnKℓ′′ µ    ℓ′′ µ λnKℓ′′ µ+1, ... λnKℓ′′ µ+ℓ′′ id    ℓ′′ id µnA [a1 . . . anA ]    , (2.31) where ℓ′′ µ and ℓ′′ id are the numbers of multiplications and intact elements in the resulting multiaction, respectively. Proposition 2.15. The arities of the polyadic field KmK ,nK , the arity nρ of the multiaction semigroup Sρ and the ℓ-shape of the polyadic compatibility (2.31) satisfy kρnA = nKℓ′′ µ + ℓ′′ id, kρ = ℓ′′ µ + ℓ′′ id. (2.32) We can exclude from (2.32) ℓ′′ ρ or ℓ′′ id and obtain nA = nK − nK − 1 kρ ℓ′′ id, nA = nK − 1 kρ ℓ′′ µ + 1, (2.33) where nK −1 kρ ℓ′′ id ≥ 1 and nK−1 kρ ℓ′′ µ ≥ 1 are integer, and the inequalities hold 1 ≤ ℓ′′ µ ≤ kρ, 0 ≤ ℓ′′ id ≤ kρ − 1, ℓ′′ µ ≤ kρ ≤ (nK − 1) ℓ′′ µ, 2 ≤ nA ≤ nK. (2.34) It follows from (2.32), that the ℓ-shape is determined by the arities and number of places kρ as ℓ′′ µ = kρ (nA − 1) nK − 1 , ℓ′′ id = kρ (nK − nA) nK − 1 . (2.35) Definition 2.16. A polyadic (“linear”) algebra over a polyadic field is the 2-set 5-operation algebraic structure AmK ,nK,mA,nA,kρ = K; A | σmK , κnK ; νmA , µnA | ρkρ , (2.36) such that the following axioms hold: 1) K; A | σmK , κnK ; νmA | ρkρ is a polyadic vector space over a polyadic field KmK ,nK ; 2) The algebra multiplication µnA and the algebra addition νmA satisfy the polyadic distributivity (1.1)–(1.3); 3) The multiplications in the algebra µnA and in the field κnK are compatible by (2.31). In the case where the algebra multiplication µnA is associative (1.4), then AmK,nK ,mA,nA,kρ is an associative polyadic algebra (for kρ = 1 see CARLSSON [1980]). If µnA is commutative, µnA [aA] = µnA [σ ◦ aA], for any polyad in algebra aA ∈ A×nA for all permutations σ ∈ Sn, where Sn is the symmetry group, then AmK ,nK,mA,nA,kρ is called a commutative polyadic algebra. As in the n-ary (semi)group theory, for polyadic algebras one can introduce special kinds of associativity and partial commutativity. If the multiplication µnA contains the identity eA (1.6) or a neutral polyad for any element, such a polyadic algebra is called unital or neutral-unital, respectively. It follows from (2.34) that: Corollary 2.17. In a polyadic (“linear”) algebra the arity of the algebra multiplication nA is less than or equal to the arity of the field multiplication nK.
  • 11. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 11 Proposition 2.18. It all the operation ℓ-shapes in (2.9), (2.16) and (2.31) coincide ℓ′′ µ = ℓ′ µ = ℓµ, ℓ′′ id = ℓ′ id = ℓid, (2.37) then, we obtain the conditions for the arities nK = mK, nρ = nA, (2.38) while mA and kρ are not connected. Proof. Use (2.14) and (2.35). Proposition 2.19. In the case of equal ℓ-shapes the multiplication and addition of the ground polyadic field (“scalars”) should coincide, while the arity nρ of the multiaction semigroup Sρ should be the same as of the algebra multiplication nA, while the arity of the algebra addition mA and number of places kρ remain arbitrary. Remark 2.20. The above ℓ-shapes (2.14), (2.21), and (2.35) are defined by a pair of integers, and therefore the arities in them are not arbitrary, but should be “quantized” in the same manner as the arities of heteromorphisms in DUPLIJ [2012]. Therefore, numerically the “quantization” rules for the ℓ-shapes (2.14), (2.21), and (2.35) coincide and given in TABLE 1. TABLE 1. “Quantization” of arity ℓ-shapes kρ ℓµ | ℓ′ µ | ℓ′′ µ ℓid | ℓ′ id | ℓ′′ id nK nρ | mK nρ | nK nA 2 1 1 3, 5, 7, . . . 2, 3, 4, . . . 3 1 2 4, 7, 10, . . . 2, 3, 4, . . . 3 2 1 4, 7, 10, . . . 3, 5, 7, . . . 4 1 3 5, 9, 13, . . . 2, 3, 4, . . . 4 2 2 3, 5, 7, . . . 2, 3, 4, . . . 4 3 1 5, 9, 13, . . . 4, 7, 10, . . . Thus, we arrive at the following Theorem 2.21 (The arity partial freedom principle). The basic two-set polyadic algebraic structures have non-free underlying operation arities which are “quantized” in such a way that their ℓ-shape is given by integers. The above definitions can be generalized, as in the binary case by considering a polyadic ring RmK ,nK instead of a polyadic field KmK ,nK . In this way a polyadic vector space becomes a polyadic module over a ring or polyadic R-module, while a polyadic algebra over a polyadic field becomes a polyadic algebra over a ring or polyadic R-algebra. All the axioms and relations between arities in the Definition 2.6 and Definition 2.16 remain the same. However, one should take into account that the ring multiplication κnK can be noncommutative, and therefore for polyadic R-modules and R- algebras it is necessary to consider many different kinds of multiactions ρkρ (all of them are described in (2.9)). For instance, in the ternary case this corresponds to left, right and central ternary modules, and tri-modules CARLSSON [1976], BAZUNOVA ET AL. [2004].
  • 12. 12 STEVEN DUPLIJ 3. MAPPINGS BETWEEN POLYADIC ALGEBRAIC STRUCTURES Let us consider DV different polyadic vector spaces over the same polyadic field KmK ,nK , as V (i) mK ,nK,m (i) V ,k (i) ρ = K; V(i) | σmK , κnK ; ν (i) m (i) V | ρ (i) k (i) ρ , i = 1, . . . , DV < ∞. (3.1) Here we define a polyadic analog of a “linear” mapping for polyadic vector spaces which “com- mutes“ with the “vector addition” and the “multiplication by scalar” (an analog of the additivity F (v + u) = F (v) + F (u), and the homogeneity of degree one F (λv) = λF (v), v, u ∈ V, λ ∈ K). Definition 3.1. A 1-place (“K-linear”) mapping between polyadic vector spaces VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ and VmK ,nK,mV ,kρ = K; V′ | σmK , κnK ; ν′ mV | ρ′ kρ over the same polyadic field KmK ,nK = K | σmK , κnK is F1 : V → V′ , such that F1 (νmV [v1, . . . , vmV ]) = ν′ mV [F1 (v1) , . . . , F1 (vmV )] , (3.2) F1  ρkρ    λ1 ... λkρ v      = ρ′ kρ    λ1 ... λkρ F1 (v)    , (3.3) where v1, . . . , vmV , v ∈ V, λ1, . . . , λkρ ∈ K. If zV is a “zero vector” in V and zV ′ is a “zero vector” in V′ (see (2.2)), then it follows from (3.2)–(3.3), that F1 (zV ) = zV ′ . The initial and final arities of νmV (“vector addition”) and the multiaction ρkρ (“multiplication by scalar”) coincide, because F1 is a 1-place mapping (a linear homomorphism). In DUPLIJ [2012] multiplace mappings and corresponding heteromorphisms were introduced. The latter allow us to change arities (mV → m′ V , kρ → k′ ρ), which is the main difference between binary and polyadic mappings. Definition 3.2. A kF -place (“K-linear”) mapping between two polyadic vector spaces VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ and VmK ,nK,mV ,kρ = K; V′ | σmK , κnK ; ν′ m′ V | ρ′ k′ ρ over the same polyadic field KmK ,nK = K | σmK , κnK is defined, if there exists FkF : V×kF → V′ , such that FkF            νmV [v1, . . . , vmV ] ... νmV vmV (ℓk µ−1), . . . vmV ℓk µ    ℓk µ vmV ℓk µ+1, ... vmV ℓk µ+ℓk id    ℓk id            = ν′ m′ V   FkF   v1 ... vkF   , . . . , FkF    vkF (m′ V −1) ... vkF m′ V       , (3.4)
  • 13. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 13 FkF                    ρkρ    λ1 ... λkρ v1    ... ρkρ    λkρ(ℓf µ−1) ... λkρℓf µ vℓf µ       ℓf µ vℓf µ+1 ... vkF    ℓf id                    = ρ′ k′ ρ    λ1 ... λk′ ρ FkF   v1 ... vkF      , (3.5) where v1, . . . , vmV , v ∈ V, λ1, . . . , λkρ ∈ K, and the four integers ℓk ρ, ℓk id, ℓf ρ, ℓf id define the ℓ-shape of the mapping. It follows from (3.4)–(3.5), that the arities satisfy kF m′ V = mV ℓk µ + ℓk id, kF = ℓk µ + ℓk id, kF = ℓf µ + ℓf id, k′ µ = kρℓf µ. (3.6) The following inequalities hold 1 ≤ ℓk µ ≤ kF , 0 ≤ ℓk id ≤ kF − 1, ℓk µ ≤ kF ≤ (mV − 1) ℓk µ, 2 ≤ m′ V ≤ mV , 2 ≤ kρ ≤ k′ ρ. (3.7) Thus, the ℓ-shape of the kF -place mapping between polyadic vector spaces is determined by ℓk µ = kF (mV − 1) mV − 1 , ℓk id = kF (mV − m′ V ) mV − 1 , ℓf µ = kρ k′ ρ , ℓf id = kF − kρ k′ ρ . (3.8) 3.1. Polyadic functionals and dual polyadic vector spaces. An important particular case of the kF - place mapping can be considered, where the final polyadic vector space coincides with the underlying field (analog of a “linear functional”). Definition 3.3. A “linear” polyadic functional (or polyadic dual vector, polyadic covector) is a kL- place mapping of a polyadic vector space VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ into its polyadic field KmK ,nK = K | σmK , κnK , such that there exists LkL : V×kL → K, and LkL            νmV [v1, . . . , vmV ] ... νmV vmV (ℓk ν−1), . . . vmV ℓk ν    ℓk ν vmV ℓk ν+1, ... vnK ℓk ν+ℓν id    ℓν id            = σmK  LkL   v1 ... vkL   , . . . , LkL   vkL(mK −1) ... vkLmK     , (3.9)
  • 14. 14 STEVEN DUPLIJ LkL                    ρkρ    λ1 ... λkρ v1    ... ρkρ    λkρ(ℓL µ −1) ... λkρℓL µ vℓL µ       ℓL µ vℓL µ +1 ... vkL    ℓL id                    = κnK  λ1, . . . , λnK−1, LkL   v1 ... vkL     , (3.10) where v1, . . . , vmV , v ∈ V , λ1, . . . , λnK ∈ K, and the integers ℓk ν, ℓν id, ℓL µ, ℓL id define the ℓ-shape of LkL . It follows from (3.4)–(3.5), that the arities satisfy kLmK = mV ℓk ν + ℓν id, kL = ℓk ν + ℓν id, kL = ℓh µ + ℓh id, nK − 1 = kρℓh µ, (3.11) and for them 1 ≤ ℓk ν ≤ kL, 0 ≤ ℓν id ≤ kL −1, ℓk ν ≤ kL ≤ (mV − 1) ℓk ν, 2 ≤ mK ≤ mV , 2 ≤ kρ ≤ nK −1. (3.12) Thus, the ℓ-shape of the polyadic functional is determined by ℓk ν = kL (mK − 1) mV − 1 , ℓν id = kL (mV − mK) mV − 1 , ℓh µ = kρ nK − 1 , ℓh id = kL − kρ nK − 1 . (3.13) In the binary case, because the dual vectors (linear functionals) take their values in the underlying field, new operations between them, such that the dual vector “addition” (+∗ ) and the “multiplication by a scalar” (•∗ ) can be naturally introduced by L(1) +∗ L(2) (v) = L(1) (v)+L(2) (v), (λ •∗ L) (v) = λ•L (v), which leads to another vector space structure, called a dual vector space. Note that operations +∗ and +, •∗ and • are different, because + and • are the operations in the underlying field K. In the polyadic case, we have more complicated arity changing formulas, and here we consider finite- dimensional spaces only. The arities of operations between dual vectors can be different from ones in the underlying polyadic field KmK nK , in general. In this way, we arrive to the following Definition 3.4. A polyadic dual vector space over a polyadic field KmK ,nK is V∗ mK ,nK,m∗ V ,k∗ ρ = K; L (i) kL | σmK , κnK ; ν∗ mL | ρ∗ kL , (3.14) and the axioms are: 1) K | σmK , κnK is a polyadic (mK, nK)-field KmK ,nK ; 2) L (i) kL | ν∗ mL , i = 1, . . . , DL is a commutative mL-ary group (which is finite, if DL < ∞); 3) The “dual vector addition” ν∗ mL is compatible with the polyadic field addition σmK by ν∗ mL L (1) kL , . . . , L (mL) kL a(kL) = σmK L (1) kL a(kL) , . . . , L (mK ) kL v(kL) , (3.15) where v(kL) =   v1 ... vkL  , v1, . . . , vkL ∈ V, and it follows that mL = mK. (3.16)
  • 15. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 15 4) The compatibility of ρ∗ kL with the “multiplication by a scalar” in the underlying polyadic field ρ∗ kL    λ1 ... λkL LkL    v(kL) = κnK λ1, . . . , λnK−1, LkL v(kL) , (3.17) and then kL = nK − 1 (3.18) 5) ρ∗ kL | composition is a nL-ary semigroup SL (similar to (2.9)) ρ∗ kL nL    λ1 ... λkL . . . ρ∗ kL    λkL(nL−1) ... λkLnL LkL    . . .    v(kL) (3.19) = ρ∗ kL    κnK [λ1, . . . λnK ] , ... κnK λnK(ℓL µ −1), . . . λnK ℓL µ    ℓL µ λnK ℓL µ +1, ... λnKℓL µ +ℓL id    ℓL id LkL    v(kL) , (3.20) where the ℓ-shape is determined by the system kLnL = nKℓL µ + ℓL id, kL = ℓL µ + ℓL id. (3.21) Using (3.18) and (3.21), we obtain the ℓ-shape as ℓL µ = nL − 1, ℓL id = nK − nL. (3.22) Corollary 3.5. The arity nL of the semigroup SL is less than or equal to the arity nK of the underlying polyadic field nL ≤ nK. 3.2. Polyadic direct sum and tensor product. The Cartesian product of DV polyadic vector spaces ×ΠmV i=1V (i) mK nKm (i) V k (i) ρ (sometimes we use the concise notation ×ΠV(i) ), i = 1, . . . , DV is given by the DV -ples (an analog of the Cartesian pair (v, u), v ∈ V(1) , u ∈ V(2) )    v(1) ... v(DV )    ≡ v(DV ) ∈ V×DV . (3.23) We introduce a polyadic generalization of the direct sum and tensor product of vector spaces by considering “linear” operations on the DV -ples (3.23). In the first case, to endow ×ΠV(i) with the structure of a vector space we need to define a new operation between the DV -ples (3.23) (similar to vector addition, but between elements from different spaces) and a rule, specifying how they are “multiplied by scalars” (analogs of (v1, v2) + (u1, u2) = (v1 + u1, v2 + u2) and λ (v1, v2) = (λv1, λv2) ). In the binary case, a formal summation is used, but it can be different from the addition in the initial vector spaces. Therefore, we can define on the set of the DV -ples (3.23) new operations χmV (“addition of vectors from different spaces”) and “multiplication by a scalar” τkρ , which does not need to coincide with the corresponding operations ν (i) m (i) V and ρ (i) k (i) ρ of the initial polyadic vector spaces V (i) mK ,nK,m (i) V ,k (i) ρ .
  • 16. 16 STEVEN DUPLIJ If all DV -ples (3.23) are of fixed length, then we can define their “addition” χmV in the standard way, if all the arities m (i) V coincide and equal the arity of the resulting vector space mV = m (1) V = . . . = m (DV ) V , (3.24) while the operations (“additions”) themselves ν (i) mV between vectors in different spaces can be still different. Thus, a new commutative mV -ary operation (“addition”) χmV of the DV -ples of the same length is defined by χmV       v (1) 1 ... v (DV ) 1    , . . . ,    v (1) mV ... v (DV ) mV       =      ν (1) mV v (1) 1 , . . . , v (1) mV ... ν (DV ) mV v (DV ) 1 , . . . , v (DV ) mV      , (3.25) where DV = mV , in general. However, it is also possible to add DV -ples of different length such that the operation (3.25) is compatible with all arities m (i) V , i = 1, . . . , mV . For instance, if mV = 3, m (1) V = m (2) V = 3, m (3) V = 2, then χ3       v (1) 1 v (2) 1 v (3) 1    ,    v (1) 2 v (2) 2 v (3) 2    ,   v (1) 3 v (2) 3      =      ν (1) 3 v (1) 1 , v (1) 2 , v (1) 3 ν (2) 3 v (2) 1 , v (2) 2 , v (2) 3 ν (3) 2 v (3) 1 , v (3) 2      . (3.26) Assertion 3.6. In the polyadic case, a direct sum of polyadic vector spaces having different arities of “vector addition” m (i) V can be defined. Let us introduce the multiaction τkρ (“multiplication by a scalar”) acting on DV -ple v(mV ) , then ρkρ    λ1 ... λkρ    v(1) ... v(DV )       =               ρ (1) k (1) ρ    λ1 ... λk (1) ρ v(1)    ... ρ (mV ) k (DV ) ρ    λ k (1) ρ +...+k (DV −1) ρ +1 ... λ k (1) ρ +...+k (DV ) ρ v(DV )                  , (3.27) where k(1) ρ + . . . + k(DV ) ρ = kρ. (3.28) Definition 3.7. A polyadic direct sum of mV polyadic vector spaces is their Cartesian product equipped with the mV -ary addition χmV and the kρ-place multiaction τkρ , satisfying (3.25) and (3.27) respectively ⊕ ΠDV i=1V (i) mK ,nK,m (i) V ,k (i) ρ = ×ΠDV i=1V (i) mK ,nK,m (i) V ,k (i) ρ | χmV , τkρ . (3.29) Let us consider another way to define a vector space structure on the DV -ples from the Carte- sian product ×ΠV(i) . Remember that in the binary case, the concept of bilinearity is used, which means “distributivity” and “multiplicativity by scalars” on each place separately in the Cartesian pair
  • 17. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 17 (v1, v2) ∈ V(1) × V(2) (as opposed to the direct sum, where these relations hold on all places simulta- neously, see (3.25) and (3.27)) such that (v1 + u1, v2) = (v1, v2) + (u1, v2) , (v1, v2 + u2) = (v1, v2) + (v1, u2) , (3.30) λ (v1, v2) = (λv1, v2) = (v1, λv2) , (3.31) respectively. If we denote the ideal corresponding to the relations (3.30)–(3.31) by B2, then the binary tensor product of the vector spaces can be defined as their Cartesian product by factoring out this ideal, as V(1) ⊗V(2) def = V(1) ×V(2) B2. Note first, that the additions and multiplications by a scalar on both sides of (3.30)–(3.31) “work” in different spaces, which sometimes can be concealed by adding the word “formal” to them. Second, all these operations have the same arity (binary ones), which need not be the case when considering polyadic structures. As in the case of the polyadic direct sum, we first define a new operation ˜χmV (“addition”) of the DV -ples of fixed length (different from χmV in (3.25)), when all the arities m (i) V coincide and equal to mV (3.24). Then, a straightforward generalization of (3.30) can be defined for mV -ples similar to the polyadic distributivity (1.1)–(1.3), as in the following mV relations      ν (1) mV v (1) 1 , . . . , v (1) mV u2 ... uDV      = ˜χmV           v (1) 1 u2 ... uDV      , . . . ,      v (1) mV u2 ... uDV           , (3.32)      u1 ν (2) mV v (2) 1 , . . . , v (2) mV ... umV      = ˜χmV           u1 v (2) 1 ... umV      , . . . ,      u1 v (2) mV ... umV           , (3.33) ...      u1 u2 ... ν (mV ) mV v (DV ) 1 , . . . , v (DV ) mV      = ˜χmV           u1 u2 ... v (DV ) 1      , . . . ,      u1 u2 ... v (DV ) mV           . (3.34) By analogy, if all k (i) ρ are equal we can define a new multiaction ˜τkρ (different from τkρ (3.27)) with the same number of places kρ = k(1) ρ = . . . = k(DV ) ρ (3.35) as the DV relations (an analog of (3.31)) ρ′ kρ    λ1 ... λkρ    v(1) ... v(DV )       =          ρ (1) kρ    λ1 ... λkρ v(1)    v(2) ... v(DV )          (3.36)
  • 18. 18 STEVEN DUPLIJ =          v(1) ρ (2) kρ    λ1 ... λkρ v(2)    ... v(DV )          (3.37) ... =          v(1) v(2) ... ρ (DV ) kρ    λ1 ... λkρ v(DV )             . (3.38) Let us denote the ideal corresponding to the relations (3.32)–(3.34), (3.36)–(3.38) by BDV . Definition 3.8. A polyadic tensor product of DV polyadic vector spaces V (i) mK ,nK,m (i) V ,k (i) ρ is their Carte- sian product equipped with the mV -ary addition ˜χmV (of DV -ples) and the kρ-place multiaction ˜τkρ , satisfying (3.32)–(3.34) and (3.36)–(3.38), respectively, by factoring out the ideal BDV ⊗ ΠmV i=1V (i) mK ,nK,m (i) V ,k (i) ρ = ×ΠmV i=1V (i) mK ,nK,m (i) V ,k (i) ρ | ˜χmV , ˜τkρ BDV . (3.39) As in the case of the polyadic direct sum, we can consider the distributivity for DV -ples of different length. In a similar example (3.26), if mV = 3, m (1) V = m (2) V = 3, m (3) V = 2, we have    ν (1) 3 v (1) 1 , v (1) 2 , v (1) 3 u2 u3    = ˜χ3     v (1) 1 u2 u3   ,   v (1) 2 u2 u3   ,   v (1) 3 u2     , (3.40)    u1 ν (2) 3 v (2) 1 , v (2) 2 , v (2) 3 u3    = ˜χ3     u1 v (2) 1 u3   ,   u1 v (2) 2 u3   ,   u1 v (2) 3     , (3.41)    u1 u2 ν (3) 2 v (3) 1 , v (3) 2    = ˜χ3     u1 u2 v (3) 1   ,   u1 u2 v (3) 2   ,   u1 u2     . (3.42) Assertion 3.9. A tensor product of polyadic vector spaces having different arities of the “vector addition” m (i) V can be defined. In the case of modules over a polyadic ring, the formulas connecting arities and ℓ-shapes similar to those above hold, while concrete properties (noncommutativity, mediality, etc.) should be taken into account.
  • 19. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 19 4. POLYADIC INNER PAIRING SPACES AND NORMS Here we introduce the next important operation: a polyadic analog of the inner product for polyadic vector spaces - a polyadic inner pairing2 . Let VmK ,nK,mV ,kρ = K; V | σmK , κnK ; νmV | ρkρ be a polyadic vector space over the polyadic field KmK ,nK (2.22). By analogy with the binary inner product, we introduce its polyadic counterpart and study its arity shape. Definition 4.1. A polyadic N-place inner pairing (an analog of the inner product) is a mapping N •|•| . . . |• : V×N → K, (4.1) satisfying the following conditions: 1) Polyadic “linearity” (2.9) (for first argument): ρkρ    λ1 ... λkρ v1    |v2| . . . |vN = κnK λ1, . . . , λkρ , v1|v2| . . . |vN . (4.2) 2) Polyadic “distributivity” (1.1)–(1.3) (on each place): νmV [v1, u1, . . . umV −1] |v2| . . . |vN = σmK [ v1|v2| . . . |vN , u1|v2| . . . |vN . . . umV −1|v2| . . . |vN ] . (4.3) If the polyadic field KmK ,nK contains the zero zK and V | mV has the zero “vector” zV (which is not always the case in the polyadic case), we have the additional axiom: 3) The polyadic inner pairing vanishes v1|v2| . . . |vN = zK, iff any of the “vectors” vanishes, ∃i ∈ 1, . . . , N, such that vi = zV . If the standard binary ordering on KmK ,nK can be defined, then the polyadic inner pairing satisfies: 4) The positivity condition N v|v| . . . |v ≥ zK, (4.4) 5) The polyadic Cauchy-Schwarz inequality (“triangle” inequality) κnK       nK N v1|v1| . . . |v1 , N v2|v2| . . . |v2 . . . N vnK |vnK | . . . |vnK       ≥ κnK   nK v1|v2| . . . |vN , v1|v2| . . . |vN . . . , v1|v2| . . . |vN   . (4.5) To make the above relations consistent, the arity shapes should be fixed. Definition 4.2. If the inner pairing is fully symmetric under permutations it is called a polyadic inner product. Proposition 4.3. The number of places in the multiaction ρkρ differs by 1 from the multiplication arity of the polyadic field nK − kρ = 1. (4.6) Proof. It follows from the polyadic “linearity” (4.2). 2 Note that this concept is different from the n-inner product spaces considered in [Misiak, et al].
  • 20. 20 STEVEN DUPLIJ Proposition 4.4. The arities of “vector addition” and “field addition” coincide mV = mK. (4.7) Proof. Implied by the polyadic “distributivity” (4.3). Proposition 4.5. The arity of the “field multiplication” is equal to the arity of the polyadic inner pairing space nK = N. (4.8) Proof. This follows from the polyadic Cauchy-Schwarz inequality (4.5). Definition 4.6. The polyadic vector space VmK ,nK,mV ,kρ equipped with the polyadic inner pairing N •|•| . . . |• : V×N → K is called a polyadic inner pairing space HmK ,nK,mV ,kρ,N . A polyadic analog of the binary norm • : V → K can be induced by the inner pairing similarly to the binary case for the inner product (we use the form v 2 = v|v ). Definition 4.7. A polyadic norm of a “vector” v in the polyadic inner pairing space HmK ,nK,mV ,kρ,N is a mapping • N : V → K, such that κnK   nK v N , v N , . . . , v N   = N v|v| . . . |v , (4.9) nK = N, (4.10) and the following axioms apply: 1) The polyadic “linearity” ρkρ    λ1 ... λkρ v    N = κnK λ1, . . . , λkρ , v N , (4.11) nK − kρ = 1. (4.12) If the polyadic field KmK ,nK contains the zero zK and V | mV has a zero “vector” zV , then: 2) The polyadic norm vanishes v N = zK, iff v = zV . If the binary ordering on V | mV can be defined, then: 3) The polyadic norm is positive v N ≥ zK. 4) The polyadic“triangle” inequality holds σmK   mK v1 N , v2 N , . . . , vN N   ≥ νmV   mV v1 N , v2 N , . . . , vN N   , (4.13) mK = mV = N. (4.14) Definition 4.8. The polyadic inner pairing space HmK ,nK ,mV ,kρ,N equipped with the polyadic norm v N is called a polyadic normed space. Recall that in the binary vector space V over the field K equipped with the inner product •|• and the norm • , one can introduce the angle between vectors v1 · v2 · cos θ = v1|v2 , where on l.h.s. there are two binary multiplications (·). Definition 4.9. A polyadic angle between N vectors v1, v2, . . . , vnK of the polyadic inner pairing space HmK ,nK,mV ,kρ,N is defined as a set of angles ϑ = {{θi} | i = 1, 2, . . . , nK − 1} satisfying κ(2) nK [ v1 N , v2 N , . . . , vnK N , cos θ1, cos θ2, . . . , cos θnK −1] = v1|v2| . . . |vnK , (4.15)
  • 21. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 21 where κ (2) nK is a long product of two nK-ary multiplications, which consists of 2 (nK − 1) + 1 terms. We will not consider the completion with respect to the above norm (to obtain a polyadic analog of Hilbert space) and corresponding limits and boundedness questions, because it will not give additional arity shapes, in which we are mostly interested here. Instead, below we turn to some applications and new general constructions which appear from the above polyadic structures. TABLE 2. The arity signature and arity shape of polyadic algebraic structures. Structures Sets Operations and arities Arity N Name N Multiplications Additions Multiactions shape Group-like polyadic algebraic structures n-ary magma (or groupoid) 1 M 1 µn : Mn → M n-ary semigroup (and monoid) 1 S 1 µn : Sn → S n-ary quasigroup (and loop) 1 Q 1 µn : Qn → Q n-ary group 1 G 1 µn : Gn → G Ring-like polyadic algebraic structures (m, n)-ary ring 1 R 2 µn : Rn → R νm : Rm → R (m, n)-ary field 1 K 2 µn : Kn → K νm : Km → K Module-like polyadic algebraic structures Module over (m, n) -ring 2 R, M 4 σn : Rn → R κm : Rm → R νmM : MmM → M ρkρ : Rkρ × M → M Vector space over (mK , nK) -field 2 K, V 4 σnK : KnK → K κmK : KmK → K νmV : VmV → V ρkρ : Kkρ × V → V (2.14) (2.21) Algebra-like polyadic algebraic structures Inner pairing space over (mK , nK) -field 2 K, V 5 σnK : KnK → K N-Form •..• : VN → K κmK : KmK → K νmV : VmV → V ρkρ : Kkρ × V → V (4.6) (4.7) (4.8) (mA, nA) -algebra over (mK , nK) -field 2 K, A 5 σnK : KnK → K µnA : An → A κmK : KmK → K νmA : AmM → A ρkρ : Kkρ × A → A (2.35) To conclude, we present the resulting TABLE 2 in which the polyadic algebraic structures are listed together with their arity shapes. APPLICATIONS 5. ELEMENTS OF POLYADIC OPERATOR THEORY Here we consider the 1-place polyadic operators T = FkF =1 (the case kF = 1 of the mapping FkF in Definition 3.2) on polyadic inner pairing spaces and structurally generalize the adjointness and involution concepts. Remark 5.1. A polyadic operator is a complicated mapping between polyadic vector spaces having nontrivial arity shape (3.4) which is actually an action on a set of “vectors”. However, only for kF = 1 it can be written in a formal way multiplicatively, as it is always done in the binary case. Recall (to fix notations and observe analogies) the informal standard introduction of the operator algebra and the adjoint operator on a binary pre-Hilbert space H (≡ HmK =2,nK=2,mV =2,kρ=1,N=2) over a binary field K (≡ KmK =2,nK=2) (having the underlying set {K; V}). For the operator norm • T : {T } → K we use (among many others) the following definition T T = inf {M ∈ K | T v ≤ M v , ∀v ∈ V} , (5.1)
  • 22. 22 STEVEN DUPLIJ which is convenient for further polyadic generalization. Bounded operators have M < ∞. If on the set of operators {T } (as 1-place mappings V → V) one defines the addition (+T ), product (◦T ) and scalar multiplication (·T ) in the standard way (T1 +T T2) (v) = T1v + T2v, (5.2) (T1 ◦T T2) (v) = T1 (T2v) , (5.3) (λ ·T T ) (v) = λ (T v) , λ ∈ K, v ∈ V, (5.4) then {T } | +T , ◦T |·T becomes an operator algebra AT (associativity and distributivity are obvious). The unity I and zero Z of AT (if they exist), satisfy Iv = v, (5.5) Zv = zV , ∀v ∈ V, (5.6) respectively, where zV ∈ V is the polyadic “zero-vector”. The connection between operators, linear functionals and inner products is given by the Riesz rep- resentation theorem. Informally, it states that in a binary pre-Hilbert space H = {K; V} a (bounded) linear functional (sesquilinear form) L : V × V → K can be uniquely represented as L (v1, v2) = T v1|v2 sym , ∀v1, v2 ∈ V, (5.7) where •|• sym : V × V → K is a (binary) inner product with standard properties and T : V → V is a bounded linear operator, such that the norms of L and T coincide. Because the linear functionals form a dual space (see Subsection 3.1), the relation (5.7) fixes the shape of its elements. The main consequence of the Riesz representation theorem is the existence of the adjoint: for any (bounded) linear operator T : V → V there exists a (unique bounded) adjoint operator T ∗ : V → V satisfying L (v1, v2) = T v1|v2 sym = v1|T ∗ v2 sym , ∀v1, v2 ∈ V, (5.8) and the norms of T and T ∗ are equal. It follows from the conjugation symmetry of the standard binary inner product, that (5.8) coincides with v1|T v2 sym = T ∗ v1|v2 sym , ∀v1, v2 ∈ V. (5.9) However, when •|• has no symmetry (permutation, conjugation, etc., see, e.g. MIGNOT [1976]), it becomes the binary (N = 2) inner pairing (4.1), the binary adjoint consists of 2 opera- tors (T ⋆12 ) = (T ⋆21 ), T ⋆ij : V → V, which should be defined by 2 equations T v1|v2 = v1|T ⋆12 v2 , (5.10) v1|T v2 = T ⋆21 v1|v2 , (5.11) where (⋆12) = (⋆21) are 2 different star operations satisfying 2 relations T ⋆12⋆21 = T , (5.12) T ⋆21⋆12 = T. (5.13) If •|• = •|• sym is symmetric, it becomes the inner product in the pre-Hilbert space H and the equations (5.12)–(5.13) coincide, while the operation (∗) = (⋆12) = (⋆21) stands for the standard involution T ∗∗ = T . (5.14)
  • 23. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 23 5.1. Multistars and polyadic adjoints. Consider now a special case of the polyadic inner pairing space (see Definition 4.6) HmK ,nK,mV ,kρ=1,N = K; V | σmK , κnK ; νmV | ρkρ=1 | N •| . . . |• (5.15) with 1-place multiaction ρkρ=1. Definition 5.2. The set of 1-place operators T : V → V together with the set of “scalars” K be- come a polyadic operator algebra AT = K; {T } | σmK , κnK ; ηmT , ωnT | θkF =1 , if the operations ηmT , ωnT , θkF =1 to define by ηmT [T1, T2, . . . , TmT ] (v) = νmV [T1v, T2v, . . . , TmT v] , (5.16) ωnT [T1, T2, . . . , TnT ] (v) = T1 (T2 . . . (TnT v)) , (5.17) θkF =1 {λ | T } (v) = ρkρ=1 {λ | T v} , ∀λ ∈ K, ∀v ∈ V. (5.18) The arity shape is fixed by Proposition 5.3. In the polyadic algebra AT the arity of the operator addition mT coincides with the “vector” addition of the inner pairing space mV , i.e. mT = mV . (5.19) Proof. This follows from (5.16). To get relations between operators we assume (as in the binary case) uniqueness: for any T1, T2 : V → V it follows from v1|v2| . . . |T1vi| . . . vN−1|vN = v1|v2| . . . |T2vi| . . . vN−1|vN , (5.20) that T1 = T2 on any place i = 1, . . . , N. First, by analogy with the binary adjoint (5.8) we define N different adjoints for each operator T . Definition 5.4. Given a polyadic operator T : V → V on the polyadic inner pairing space HmK ,nK,mV ,kρ=1,N we define a polyadic adjoint as the set {T ⋆ij } of N operators T ⋆ij satisfying the following N equations T v1|v2|v3| . . . |vN = v1|T ⋆12 v2|v3| . . . |vN , v1|T v2|v3| . . . |vN = v1|v2|T ⋆23 v3| . . . |vN , ... v1|v2|v3| . . . T vN−1|vN = v1|v2|v3| . . . |T ⋆N−1,N vN , v1|v2|v3| . . . vN−1|T vN = T ⋆N,1 v1|v2|v3| . . . |vN , vi ∈ V. (5.21) In what follows, for the composition we will use the notation (T ⋆ij )⋆kl... ≡ T ⋆ij ⋆kl... . We have from (5.21) the N relations T ⋆12⋆23⋆34...⋆N−1,N ⋆N,1 = T, T ⋆23⋆34...⋆N−1,N ⋆N,1⋆12 = T, ... T ⋆N,1⋆12⋆23⋆34...⋆N−1,N = T, (5.22) which are called multistar cycles. Definition 5.5. We call the set of adjoint mappings (•⋆ij ) : T → T ⋆ij a polyadic involution, if they satisfy the multistar cycles (5.22).
  • 24. 24 STEVEN DUPLIJ If the inner pairing •| . . . |• has more than two places N ≥ 3, we have some additional structural issues, which do not exist in the binary case. First, we observe that the set of the adjointness relations (5.21) can be described in the framework of the associativity quiver approach introduced in DUPLIJ [2012] for polyadic representations. That is, for general N ≥ 3 in addition to (5.21) which corresponds to the so called Post-like associativity quiver (they will be called the Post-like adjointness relations), there also exist other sets. It is cum- bersome to write additional general formulas like (5.21) for other non-Post-like cases, while instead we give a clear example for N = 4. Example 5.6. The polyadic adjointness relations for N = 4 consist of the sets corresponding to different associativity quivers 1) Post-like adjointness relations T v1|v2|v3|v4 = v1|T ⋆12 v2|v3|v4 , v1|T v2|v3|v4 = v1|v2|T ⋆23 v3|v4 , v1|v2|T v3|v4 = v1|v2|v3|T ⋆34 v4 , v1|v2|v3|T v4 = T ⋆41 v1|v2|v3|v4 , 2) Non-Post-like adjointness relations T v1|v2|v3|v4 = v1|v2|v3|T ⋆14 v4 , v1|v2|v3|T v4 = v1|v2|T ⋆43 v3|v4 , v1|v2|T v3|v4 = v1|T ⋆32 v2|v3|v4 , v1|T v2|v3|v4 = T ⋆21 v1|v2|v3|v4 , (5.23) and the corresponding multistar cycles 1) Post-like multistar cycles T ⋆12⋆23⋆34⋆41 = T , T ⋆23⋆34⋆41⋆12 = T , T ⋆34⋆41⋆12⋆23 = T , T ⋆41⋆12⋆23⋆34 = T , 2) Non-Post-like multistar cycles T ⋆14⋆43⋆32⋆21 = T, T ⋆43⋆32⋆21⋆14 = T, T ⋆32⋆21⋆14⋆43 = T, T ⋆21⋆14⋆43⋆32 = T. (5.24) Thus, if the inner pairing has no symmetry, then both the Post-like and non-Post-like adjoints and corresponding multistar involutions are different. Second, in the case N ≥ 3 any symmetry of the multiplace inner pairing restricts the polyadic adjoint sets and multistar involutions considerably. Theorem 5.7. If the inner pairing with N ≥ 3 has the full permutation symmetry v1|v2| . . . |vN = σv1|σv2| . . . |σvN , ∀σ ∈ SN , (5.25) where SN is the symmetric group of N elements, then: 1) All the multistars coincide (⋆ij) = (⋆kl) := (∗) for any allowed i, j, k, l = 1, . . . , N; 2) All the operators are self-adjoint T = T ∗ . Proof. 1) In each adjointness relation from (5.21) we place the operator T on the l.h.s. to the first position and its multistar adjoint T ⋆ij to the second position, using the full permutation symmetry, which together with (5.20) gives the equality of all multistar operations. 2) We place the operator T on the l.h.s. to the first position and apply the derivation of the involution in the binary case to increasing cycles of size i ≤ N recursively, that is: For i = 2 T v1|v2|v3| . . . |vN = v1|T ∗ v2|v3| . . . |vN = T ∗ v2|v1|v3| . . . |vN = v2|T ∗∗ v1|v3| . . . |vN = T ∗∗ v1|v2|v3| . . . |vN , (5.26) then, using (5.20) we get T = T ∗∗ , (5.27) as in the standard binary case. However, for N ≥ 3 we have N higher cycles in addition.
  • 25. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 25 For i = 3 T v1|v2|v3| . . . |vN = v1|T ∗ v2|v3| . . . |vN = T ∗ v2|v3|v1| . . . |vN = v2|T ∗∗ v3|v1| . . . |vN = T ∗∗ v3|v1|v2| . . . |vN = v3|T ∗∗∗ v1|v2| . . . |vN = T ∗∗∗ v1|v2|v3| . . . |vN , (5.28) which together with (5.20) gives T = T ∗∗∗ , (5.29) and after using (5.27) T = T ∗ . (5.30) Similarly, for an arbitrary length of the cycle i we obtain T = T i ∗ ∗ . . . ∗, which should be valid for each cycle recursively with i = 2, 3, . . . , N. Therefore, for any N ≥ 3 all the operators T are self-adjoint (5.30), while N = 2 is an exceptional case, when we have T = T ∗∗ (5.27) only. Now we show that imposing a partial symmetry on the polyadic inner pairing will give more inter- esting properties to the adjoint operators. Recall, that one of possible binary commutativity general- izations of (semi)groups to the polyadic case is the semicommutativity concept, when in the multipli- cation only the first and last elements are exchanged. Similarly, we introduce Definition 5.8. The polyadic inner pairing is called semicommutative, if v1|v2|v3| . . . |vN = vN |v2|v3| . . . |v1 , vi ∈ V. (5.31) Proposition 5.9. If the polyadic inner pairing is semicommutative, then for any operator T (satis- fying Post-like adjointness (5.21)) the last multistar operation (⋆N,1) is a binary involution and is a composition of all the previous multistars T ⋆N,1 = T ⋆12⋆23⋆34...⋆N−1,N , (5.32) T ⋆N,1⋆N,1 = T . (5.33) Proof. It follows from (5.21) and (5.31), that v1|v2|v3| . . . |T vN = T vN |v2|v3| . . . |v1 = vN |v2|v3| . . . |T ⋆12⋆23⋆34...⋆N−1,N v1 (5.34) = T ⋆12⋆23⋆34...⋆N−1,N v1|v2|v3| . . . |vN = T ⋆N,1 v1|v2|v3| . . . |vN , (5.35) which using (5.20) gives (5.32), (5.33) follows from the first multistar cocycle in (5.22). The adjointness relations (5.21) (of all kinds) together with (5.18) and (5.19) allows us to fix the arity shape of the polyadic operator algebra AT . We will assume that the arity of the operator multi- plication in AT coincides with the number of places of the inner pairing N (4.1) nT = N, (5.36) because it is in agreement with (5.21). Thus, the arity shape of the polyadic operator algebra becomes AT = K; {T } | σmK , κnK ; ηmT =mV , ωnT =N | θkF =kρ=1 , (5.37) Definition 5.10. We call the operator algebra AT which has the arity nT = N a nonderived polyadic operator algebra. Let us investigate some structural properties of AT and types of polyadic operators.
  • 26. 26 STEVEN DUPLIJ Remark 5.11. We can only define, but not derive, as in the binary case, the action of any multistar (⋆ij) on the product of operators, because in the nonderived nT -ary algebra we have a fixed number of operators in a product and sum, that is ℓ′ (nT − 1)+1 and ℓ′′ (mT − 1)+1, correspondingly, where ℓ′ is the number of nT -ary multiplications and ℓ′ is the number of mT -ary additions. Therefore, we cannot transfer (one at a time) all the polyadic operators from one place in the inner pairing to another place, as is done in the standard proof in the binary case. Taking this into account, as well as consistency under the multistar cycles (5.22), we arrive at Definition 5.12. The fixed multistar operation acts on the ℓ = 1 product of nT polyadic operators, depending on the sequential number of the multistar (⋆ij) (for the Post-like adjointness relations (5.21)) sij := i + j − 1 2 , if 3 ≤ i + j ≤ 2N − 1 N, if i j = N, sij = 1, 2, . . . , N − 1, N, (5.38) in the following way (ωnT [T1, T2, . . . , TnT −1, TnT ])⋆ij = ωnT T ⋆ij nT , T ⋆ij nT −1, . . . , T ⋆ij 2 , T ⋆ij 1 , if sij is odd, ωnT T ⋆ij 1 , T ⋆ij 2 , . . . , T ⋆ij nT −1, T ⋆ij nT , if sij is even. (5.39) A rule similar to (5.39) holds also for non-Post-like adjointness relations, but their concrete form depends of the corresponding non-Post-like associative quiver. Sometimes, to shorten notation, it will be more convenient to mark a multistar by the sequential number (5.38), such that (⋆ij) ⇒ ⋆sij , e.g. (⋆23) ⇒ (⋆2), (⋆N,1) ⇒ (⋆N ), etc. Also, in examples, for the ternary multiplication we will use the square brackets without the name of operation, if it is clear from the context, e.g. ω3 [T1, T2, T3] ⇒ [T1, T2, T3], etc. Example 5.13. In the lowest ternary case N = 3 we have 1) Post-like adjointness relations T v1|v2|v3 = v1|T ⋆1 v2|v3 , v1|T v2|v3 = v1|v2|T ⋆2 v3 , v1|v2|T v3 = T ⋆3 v1|v2|v3 , 2) Non-Post-like adjointness relations T v1|v2|v3 = v1|v2|T ⋆3 v3 , v1|v2|T v3 = v1|T ⋆2 v2|v3 , v1|T v2|v3 = T ⋆1 v1|v2|v3 , (5.40) and the corresponding multistar cycles 1) Post-like multistar cycles T ⋆1⋆2⋆3 = T , T ⋆2⋆3⋆1 = T , T ⋆3⋆1⋆2 = T , 2) Non-Post-like multistar cycles T ⋆3⋆2⋆1 = T, T ⋆2⋆1⋆3 = T, T ⋆1⋆3⋆2 = T. (5.41) Using (5.39) we obtain the following ternary conjugation rules ([T1, T2, T3])⋆1 = [T ⋆1 3 , T ⋆1 2 , T ⋆1 1 ] , (5.42) ([T1, T2, T3])⋆2 = [T ⋆2 1 , T ⋆2 2 , T ⋆2 3 ] , (5.43) ([T1, T2, T3])⋆3 = [T ⋆3 3 , T ⋆3 2 , T ⋆3 1 ] , (5.44) which are common for both Post-like and non-Post-like adjointness relations (5.40). Definition 5.14. A polyadic operator T is called self-adjoint, if all multistar operations are identities, i.e. (⋆ij) = id, ∀i, j.
  • 27. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 27 5.2. Polyadic isometry and projection. Now we introduce polyadic analogs for the important types of operators: isometry, unitary, and (orthogonal) projection. Taking into account Remark 5.11, we again cannot move operators singly, and instead of proving the operator relations, as it is usually done in the binary case, we can only exploit some mnemonic rules to define the corresponding relations between polyadic operators. If the polyadic operator algebra AT contains a unit I and zero Z (see (5.5)–(5.6)) we define the conditions of polyadic isometry and orthogonality: Definition 5.15. A polyadic operator T is called a polyadic isometry, if it preserves the polyadic inner pairing T v1|T v2|T v3| . . . |T vN = v1|v2|v3| . . . |vN , (5.45) and satisfies ωnT [T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ] = I, + (N − 1) cycle permutations of multistars in the first (N − 1) terms. (5.46) Remark 5.16. If the multiplication in AT is derived and all multistars are equal, then the polyadic isometry operators satisfy some kind of N-regularity DUPLIJ AND MARCINEK [2002] or regular N-cocycle condition DUPLIJ AND MARCINEK [2001]. Proposition 5.17. The polyadic isometry operator T preserves the polyadic norm T v N = v N , ∀v ∈ V. (5.47) Proof. It follows from (4.9) and (5.45), that κnK   nK T v N , T v N , . . . , T v N   = κnK   nK v N , v N , . . . , v N   , (5.48) which gives (5.47), when nK = N. Definition 5.18. If for N polyadic operators Ti we have T1v1|T2v2|T3v3| . . . |TN vN = zK, ∀vi ∈ V, (5.49) where zK ∈ V is the zero of the underlying polyadic field KmK ,nK , then we say that Ti are (polyadi- cally) orthogonal, and they satisfy ωnT T ⋆N−1,N 1 , T ⋆N−2,N−1⋆N−1,N 2 , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N 3 , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N N−1 , TN = Z, (5.50) + (N − 1) cycle permutations of multistars in the first (N − 1) terms. (5.51) The polyadic analog of projection is given by Definition 5.19. If a polyadic operator P ∈ AT satisfies the polyadic idempotency condition ωnT   nT P , P , . . . P   = P, (5.52) then it is called a polyadic projection. By analogy with the binary case, polyadic projections can be constructed from polyadic isometry operators in a natural way.
  • 28. 28 STEVEN DUPLIJ Proposition 5.20. If T ∈ AT is a polyadic isometry, then P (1) T = ωnT [T , T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N ] , + (N − 1) cycle permutations of multistars in the last (N − 1) terms. (5.53) are the corresponding polyadic projections P (k) T , k = 1, . . . , N, satisfying (5.52). Definition 5.21. A polyadic operator T ∈ AT is called normal, if ωnT [T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ] = ωnT [T , T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N ] , + (N − 1) cycle permutations of multistars in the (N − 1) terms. (5.54) Proof. Insert (5.53) into (5.52) and use (5.46) together with nT -ary associativity. Definition 5.22. If all the polyadic projections (5.53) are equal to unity P (k) T = I, then the corre- sponding polyadic isometry operator T is called a polyadic unitary operator. It can be shown, that each polyadic unitary operator is querable (“polyadically invertible”), such that it has a querelement in AT . 5.3. Towards polyadic analog of C∗ -algebras. Let us, first, generalize the operator binary norm (5.1) to the polyadic case. This can be done, provided that a binary ordering on the underlying polyadic field KmK ,nK can be introduced. Definition 5.23. The polyadic operator norm • T : {T } → K is defined by T T = inf    M ∈ K | T v N ≤ µnK   nK −1 M, . . . , M, v N   , ∀v ∈ V    , (5.55) where • N is the polyadic norm in the inner pairing space HmK ,nK,mV ,kρ=1,N and µnK is the nK-ary multiplication in KmK ,nK . Definition 5.24. The polyadic operator norm is called submultiplicative, if ωnT [T1, T2, . . . , TnT ] T ≤ µnK [ T1 T , T2 T , . . . , TnK T ] , (5.56) nT = nK. (5.57) Definition 5.25. The polyadic operator norm is called subadditive, if ηmT [T1, T2, . . . , TnT ] T ≤ νmK [ T1 T , T2 T , . . . , TmK T ] , (5.58) mT = mK. (5.59) By analogy with the binary case, we have Definition 5.26. The polyadic operator algebra AT equipped with the submultiplicative norm • T a polyadic Banach algebra of operators BT . The connection between the polyadic norms of operators and their polyadic adjoints is given by Proposition 5.27. For polyadic operators in the inner pairing space HmK ,nK,mV ,kρ=1,N
  • 29. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 29 1) The following N multi-C∗ -relations ωnT [T ⋆N−1,N , T ⋆N−2,N−1⋆N−1,N , . . . T ⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ⋆12⋆23⋆34...⋆N−2,N−1⋆N−1,N , T ] N = µnK   nK T T , T T , . . . , T T   , + (N − 1) cycle permutations of (N − 1) terms with multistars, (5.60) take place, if nT = nK. 2) The polyadic norms of operator and its all adjoints coincide T ⋆i,j T = T T , ∀i, j ∈ 1, . . . , N. (5.61) Proof. Both statements follow from (5.21) and the definition of the polyadic operator norm (5.55). Therefore, we arrive to Definition 5.28. The operator Banach algebra BT satisfying the multi-C∗ -relations is called a polyadic operator multi-C∗ -algebra. The first example of a multi-C∗ -algebra (as in the binary case) can be constructed from one isometry operator (see Definition 5.15). Definition 5.29. A polyadic algebra generated by one isometry operator T satisfying (5.46) on the inner pairing space HmK ,nK ,mV ,kρ=1,N represents a polyadic Toeplitz algebra TmT ,nT and has the arity shape mT = mV , nT = N. Example 5.30. The ternary Toeplitz algebra T3,3 is represented by the operator T and relations [T ⋆1 , T ⋆3⋆1 , T ] = I, [T ⋆2 , T ⋆1⋆2 , T ] = I, [T ⋆3 , T ⋆2⋆3 , T ] = I. (5.62) Example 5.31. If the inner pairing is semicommutative (5.31), then (⋆3) can be eliminated by T ⋆3 = T ⋆1⋆2 , (5.63) T ⋆3⋆3 = T, (5.64) and the corresponding relations representing T3,3 become [T ⋆1 , T ⋆1 , T ] = I, [T ⋆2 , T ⋆1⋆2 , T ] = I, [T ⋆1⋆2 , T ⋆2 , T ] = I. (5.65) Let us consider M polyadic operators T1T2 . . . TM ∈ BT and the related partial (in the usual sense) isometries (5.52) which are mutually orthogonal (5.50). In the binary case, the algebra generated by M operators, such that the sum of the related orthogonal partial projections is unity, represents the Cuntz algebra OM CUNTZ [1977]. Definition 5.32. A polyadic algebra generated by M polyadic isometric operators T1T2 . . . TM ∈ BT satisfying η(ℓa) mT P (k) T1 , P (k) T2 . . . P (k) TM = I, k = 1, . . . , N, (5.66) where P (k) Ti are given by (5.53)and η (ℓa) mT is a “long polyadic addition” (5.16), represents a polyadic Cuntz algebra pOM|mT ,nT , which has the arity shape M = ℓa (mT − 1) + 1, (5.67)
  • 30. 30 STEVEN DUPLIJ where ℓa is number of “mT -ary additions”. Below we use the same notations, as in Example 5.13, also the ternary addition will be denoted by (+3) as follows: η3 [T1, T2, T3] ≡ T1 +3 T2 +3 T3. Example 5.33. In the ternary case mT = nT = 3 and one ternary addition ℓa = 1, we have M = 3 mutually orthogonal isometries T1, T2, T3 ∈ BT and N = 3 multistars (⋆i). In case of the Post-like multistar cocycles (5.41) they satisfy Isometry conditions [T ⋆1 i , T ⋆3⋆1 i , Ti] = I, [T ⋆2 i , T ⋆1⋆2 i , Ti] = I, [T ⋆3 i , T ⋆2⋆3 i , Ti] = I, i = 1, 2, 3, Orthogonality conditions T ⋆1 i , T ⋆3⋆1 j , Tk = Z, T ⋆2 i , T ⋆1⋆2 j , Tk = Z, T ⋆3 i , T ⋆2⋆3 j , Tk = Z, i, j, k = 1, 2, 3, i = j = k, (5.68) and the (sum of projections) relations [T1, T ⋆1 1 , T ⋆3⋆1 1 ] +3 [T2, T ⋆1 2 , T ⋆3⋆1 2 ] +3 [T3, T ⋆1 3 , T ⋆3⋆1 3 ] = I, (5.69) [T1, T ⋆2 1 , T ⋆1⋆2 1 ] +3 [T2, T ⋆2 2 , T ⋆1⋆2 2 ] +3 [T3, T ⋆2 3 , T ⋆1⋆2 3 ] = I, (5.70) [T1, T ⋆3 1 , T ⋆2⋆3 1 ] +3 [T2, T ⋆3 2 , T ⋆2⋆3 2 ] +3 [T3, T ⋆3 3 , T ⋆2⋆3 3 ] = I, (5.71) which represent the ternary Cuntz algebra pO3|3,3. Example 5.34. In the case where the inner pairing is semicommutative (5.31), we can eliminate the multistar (⋆3) by (5.63) and represent the two-multistar ternary analog of the Cuntz algebra pO3|3,3 by [T ⋆1 i , T ⋆2 i , Ti] = I, [T ⋆2 i , T ⋆1⋆2 i , Ti] = I, [T ⋆1⋆2 i , T ⋆2 i , Ti] = I, i = 1, 2, 3, T ⋆1 i , T ⋆2 j , Tk = Z, T ⋆1 i , T ⋆1⋆2 j , Tk = Z, T ⋆1⋆2 i , T ⋆2 j , Tk = Z, i, j, k = 1, 2, 3, i = j = k, (5.72) [T1, T ⋆1 1 , T ⋆1 1 ] +3 [T2, T ⋆1 2 , T ⋆1 2 ] +3 [T3, T ⋆1 3 , T ⋆1 3 ] = I, (5.73) [T1, T ⋆2 1 , T ⋆1⋆2 1 ] +3 [T2, T ⋆2 2 , T ⋆1⋆2 2 ] +3 [T3, T ⋆2 3 , T ⋆1⋆2 3 ] = I, (5.74) [T1, T ⋆1⋆2 1 , T ⋆2 1 ] +3 [T2, T ⋆1⋆2 2 , T ⋆2 2 ] +3 [T3, T ⋆1⋆2 3 , T ⋆2 3 ] = I. (5.75) 6. CONGRUENCE CLASSES AS POLYADIC RINGS Here we will show that the inner structure of the residue classes (congruence classes) over integers is naturally described by polyadic rings CELAKOSKI [1977], CROMBEZ [1972], LEESON AND BUTSON [1980], and then study some special properties of them. Denote a residue class (congruence class) of an integer a, modulo b by3 [[a]]b = {{a + bk} | k ∈ Z, a ∈ Z+, b ∈ N, 0 ≤ a ≤ b − 1} . (6.1) A representative element of the class [[a]]b will be denoted by xk = x (a,b) k = a + bk. Here we do not consider the addition and multiplication of the residue classes (congruence classes). Instead, we consider the fixed congruence class [[a]]b, and note that, for arbitrary a and b, it is not closed under binary operations. However, it can be closed with respect to polyadic operations. 3 We use for the residue class the notation [[a]]b , because the standard notations by one square bracket [a]b or ¯ab are busy by the n-ary operations and querelements, respectively.
  • 31. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 31 6.1. Polyadic ring on integers. Let us introduce the m-ary addition and n-ary multiplication of representatives of the fixed congruence class [[a]]b by νm [xk1 , xk2 , . . . , xkm ] = xk1 + xk2 + . . . + xkm , (6.2) µn [xk1 , xk2 , . . . , xkn ] = xk1 xk2 . . . xkn , xki ∈ [[a]]b , ki ∈ Z, (6.3) where on the r.h.s. the operations are the ordinary binary addition and binary multiplication in Z. Remark 6.1. The polyadic operations (6.2)–(6.3) are not derived (see, e.g., GŁAZEK AND MICHALSKI [1984], MICHALSKI [1988]), because on the set {xki } one cannot define the binary semigroup struc- ture with respect to ordinary addition and multiplication. Derived polyadic rings which consist of the repeated binary sums and binary products were considered in LEESON AND BUTSON [1980]. Lemma 6.2. In case (m − 1) a b = I(m) (a, b) = I = integer (6.4) the algebraic structure [[a]]b | νm is a commutative m-ary group. Proof. The closure of the operation (6.2) can be written as xk1 + xk2 + . . . + xkm = xk0 , or ma + b (k1 + k2 + . . . + km) = a+bk0, and then k0 = (m − 1) a/b+(k1 + k2 + . . . + km), from (6.4). The (total) associativity and commutativity of νm follows from those of the addition in the binary Z. Each element xk has its unique querelement ˜x = x˜k determined by the equation (m − 1) xk + x˜k = xk, which (uniquely, for any k ∈ Z) gives ˜k = bk (2 − m) − (m − 1) a b . (6.5) Thus, each element is “querable” (polyadic invertible), and so [[a]]b | νm is a m-ary group. Example 6.3. For a = 2, b = 7 we have 8-ary group, and the querelement of xk is ˜x = x(−2−12k). Proposition 6.4. The m-ary commutative group [[a]]b | νm : 1) has an infinite number of neutral sequences for each element; 2) if a = 0, it has no “unit” (which is actually zero, because νm plays the role of “addition”); 3) in case of the zero congruence class [[0]]b the zero is xk = 0. Proof. 1) The (additive) neutral sequence ˜nm−1 of the length (m − 1) is defined by νm [˜nm−1, xk] = xk. Using (6.2), we take ˜nm−1 = xk1 + xk2 + . . . + xkm−1 = 0 and obtain the equation (m − 1) a + b (k1 + k2 + . . . + km−1) = 0. (6.6) Because of (6.4), we obtain k1 + k2 + . . . + km−1 = −I(m) (a, b) , (6.7) and so there is an infinite number of sums satisfying this condition. 2) The polyadic “unit”/zero z = xk0 = a + bk0 satisfies νm m−1 z, z, . . . , z, xk = xk for all xk ∈ [[a]]b (the neutral sequence ˜nm−1 consists of one element z only), which gives (m − 1) (a + bk0) = 0 having no solutions with a = 0, since a < b. 3) In the case a = 0, the only solution is z = xk=0 = 0. Example 6.5. In case a = 1, b = 2 we have m = 3 and I(3) (1, 2) = 1, and so from (6.6) we get k1 + k2 = −1, thus the infinite number of neutral sequences consists of 2 elements ˜n2 = xk + x−1−k, with arbitrary k ∈ Z.
  • 32. 32 STEVEN DUPLIJ Lemma 6.6. If an − a b = J(n) (a, b) = J = integer, (6.8) then [[a]]b | µn is a commutative n-ary semigroup. Proof. It follows from (6.3), that the closeness of the operation µn is xk1 xk2 . . . xkn = xk0 , which can be written as an + b (integer) = a + bk0 leading to (6.8). The (total) associativity and commutativity of µn follows from those of the multiplication in Z. Definition 6.7. A unique pair of integers (I, J) is called a (polyadic) shape invariants of the congru- ence class [[a]]b. Theorem 6.8. The algebraic structure of the fixed congruence class [[a]]b is a polyadic (m, n)-ring R[a,b] m,n = [[a]]b | νm, µn , (6.9) where the arities m and n are minimal positive integers (more or equal 2), for which the congruences ma ≡ a (mod b) , (6.10) an ≡ a (mod b) (6.11) take place simultaneously, fixating its polyadic shape invariants (I, J). Proof. By Lemma 6.2, 6.6 the set [[a]]b is a m-ary group with respect to “m-ary addition” νm and a n- ary semigroup with respect to “n-ary multiplication” µn, while the polyadic distributivity (1.1)–(1.3) follows from (6.2) and (6.3) and the binary distributivity in Z. Remark 6.9. For a fixed b ≥ 2 there are b congruence classes [[a]]b, 0 ≤ a ≤ b − 1, and therefore exactly b corresponding polyadic (m, n)-rings R [a,b] m,n, each of them is infinite-dimensional. Corollary 6.10. In case gcd (a, b) = 1 and b is prime, there exists the solution n = b. Proof. Follows from (6.11) and Fermat’s little theorem. Remark 6.11. We exclude from consideration the zero congruence class [[0]]b, because the arities of operations νm and µn cannot be fixed up by (6.10)–(6.11) becoming identities for any m and n. Since the arities are uncertain, their minimal values can be chosen m = n = 2, and therefore, it follows from (6.2) and (6.3), R [0,b] 2,2 = Z. Thus, in what follows we always imply that a = 0 (without using a special notation, e.g. R∗ , etc.). In TABLE 3 we present the allowed (by (6.10)–(6.11)) arities of the polyadic ring R [a,b] m,n and the corresponding polyadic shape invariants (I, J) for b ≤ 10. Let us investigate the properties of R [a,b] m,n in more detail. First, we consider equal arity polyadic rings and find the relation between the corresponding congruence classes. Proposition 6.12. The residue (congruence) classes [[a]]b and [[a′ ]]b′ which are described by the polyadic rings of the same arities R [a,b] m,n and R [a′,b′] m,n are related by b′ I′ a′ = bI a , (6.12) a′ + b′ J′ = (a + bJ)loga a′ . (6.13) Proof. Follows from (6.4) and (6.8). For instance, in TABLE 3 the congruence classes [[2]]5, [[3]]5, [[2]]10, and [[8]]10 are (6, 5)-rings. If, in addition, a = a′ , then the polyadic shapes satisfy I J = I′ J′ . (6.14)
  • 33. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 33 6.2. Limiting cases. The limiting cases a ≡ ±1 (mod b) are described by Corollary 6.13. The polyadic ring of the fixed congruence class [[a]]b is: 1) multiplicative binary, if a = 1; 2) multiplicative ternary, if a = b − 1; 3) additive (b + 1)-ary in both cases. That is, the limiting cases contain the rings R [1,b] b+1,2 and R [b−1,b] b+1,3 , respectively. They correspond to the first row and the main diagonal of TABLE 3. Their intersection consists of the (3, 2)-ring R [1,2] 3,2 . Definition 6.14. The congruence classes [[1]]b and [[b − 1]]b are called the limiting classes, and the corresponding polyadic rings are named the limiting polyadic rings of a fixed congruence class. Proposition 6.15. In the limiting cases a = 1 and a = b − 1 the n-ary semigroup [[a]]b | µn : 1) has the neutral sequences of the form ¯nn−1 = xk1 xk2 . . . xkn−1 = 1, where xki = ±1; 2) has a) the unit e = xk=1 = 1, for the limiting class [[1]]b, b) the unit e− = xk=−1 = −1, if n is odd, for [[b − 1]]b, c) the class [[1]]2 contains both polyadic units e and e− ; 3) has the set of “querable” (polyadic invertible) elements which consist of ¯x = x¯k = ±1; 4) has in the “intersecting” case a = 1, b = 2 and n = 2 the binary subgroup Z2 = {1, −1}, while other elements have no inverses. Proof. 1) The (multiplicative) neutral sequence ¯nn−1 of length (n − 1) is defined by µn [¯nn−1, xk] = xk. It follows from (6.3) and cancellativity in Z, that ¯nn−1 = xk1 xk2 . . . xkn−1 = 1 which is (a + bk1) (a + bk2) . . . (a + bkn−1) = 1. (6.15) The solution of this equation in integers is the following: a) all multipliers are a + bki = 1, i = 1, . . . , n − 1; b) an even number of multipliers can be a + bki = −1, while the others are 1. 2) If the polyadic unit e = xk1 = a + bk1 exists, it should satisfy µm n−1 e, e, . . . , e, xk = xk ∀xk ∈ [[a]]b | µn , such that the neutral sequence ¯nn−1 consists of one element e only, and this leads to (a + bk1)n−1 = 1. For any n this equation has the solution a + bk1 = 1, which uniquely gives a = 1 and k1 = 0, thus e = xk1=0 = 1. If n is odd, then there exists a “negative unit” e− = xk1=−1 = −1, such that a + bk1 = −1, which can be uniquely solved by k1 = −1 and a = b − 1. The neutral sequence becomes ¯nn−1 = n−1 e− , e− , . . . , e− = 1, as a product of an even number of e− = −1. The intersection of limiting classes consists of one class [[1]]2, and therefore it contains both polyadic units e and e− . 3) An element xk in [[a]]b | µn is “querable”, if there exists its querelement ¯x = x¯k such that µn n−1 xk, xk, . . . , xk, ¯x = xk. Using (6.3) and the cancellativity in Z, we obtain the equa- tion (a + bk)n−2 a + b¯k = 1, which in integers has 2 solutions: a) (a + bk)n−2 = 1 and a + b¯k = 1, the last relation fixes up the class [[1]]b, and the arity of multiplication n = 2, and therefore the first relation is valid for all elements in the class, each of them has the same querelement ¯x = 1. This means that all elements in [[1]]b are “querable”, but only one element x = 1 has an inverse, which is also 1; b) (a + bk)n−2 = −1 and a + b¯k = −1. The second relation fixes the class [[b − 1]]b, and from the first relation we conclude that the arity n should be odd. In this case only one element −1 is “querable”, which has ¯x = −1, as a querelement. 4) The “intersecting” class [[1]]2 contains 2 “querable” elements ±1 which coincide with their inverses, which means that {+1, −1} is a binary subgroup (that is Z2) of the binary semigroup [[1]]2 | µ2 .
  • 34. 34 STEVEN DUPLIJ Corollary 6.16. In the non-limiting cases a = 1, b − 1, the n-ary semigroup [[a]]b | µn contains no “querable” (polyadic invertible) elements at all. Proof. It follows from (a + bk) = ±1 for any k ∈ Z or a = ±1 (mod b). TABLE 3. The polyadic ring R Z(a,b) m,n of the fixed residue class [[a]]b: arity shape. a b 2 3 4 5 6 7 8 9 10 1 m = 3 n = 2 I = 1 J = 0 m = 4 n = 2 I = 1 J = 0 m = 5 n = 2 I = 1 J = 0 m = 6 n = 2 I = 1 J = 0 m = 7 n = 2 I = 1 J = 0 m = 8 n = 2 I = 1 J = 0 m = 9 n = 2 I = 1 J = 0 m = 10 n = 2 I = 1 J = 0 m = 11 n = 2 I = 1 J = 0 2 m = 4 n = 3 I = 2 J = 2 m = 6 n = 5 I = 2 J = 6 m = 4 n = 3 I = 1 J = 1 m = 8 n = 4 I = 2 J = 2 m = 10 n = 7 I = 2 J = 14 m = 6 n = 5 I = 1 J = 3 3 m = 5 n = 3 I = 3 J = 6 m = 6 n = 5 I = 3 J = 48 m = 3 n = 2 I = 1 J = 1 m = 8 n = 7 I = 3 J = 312 m = 9 n = 3 I = 3 J = 3 m = 11 n = 5 I = 3 J = 24 4 m = 6 n = 3 I = 4 J = 12 m = 4 n = 2 I = 2 J = 2 m = 8 n = 4 I = 4 J = 36 m = 10 n = 4 I = 4 J = 28 m = 6 n = 3 I = 2 J = 6 5 m = 7 n = 3 I = 5 J = 20 m = 8 n = 7 I = 5 J = 11160 m = 9 n = 3 I = 5 J = 15 m = 10 n = 7 I = 5 J = 8680 m = 3 n = 2 I = 1 J = 2 6 m = 8 n = 3 I = 6 J = 30 m = 6 n = 2 I = 3 J = 3 7 m = 9 n = 3 I = 7 J = 42 m = 10 n = 4 I = 7 J = 266 m = 11 n = 5 I = 7 J = 1680 8 m = 10 n = 3 I = 8 J = 56 m = 6 n = 5 I = 4 J = 3276 9 m = 11 n = 3 I = 9 J = 72 Based on the above statements, consider in the properties of the polyadic rings R [a,b] m,n (a = 0) describing non-zero congruence classes (see Remark 6.11). Definition 6.17. The infinite set of representatives of the congruence (residue) class [[a]]b having fixed arities and form the (m, n)-ring R [a,b] m,n is called the set of (polyadic) (m, n)-integers (numbers) and denoted Z(m,n). Just obviously, for ordinary integers Z = Z(2,2), and they form the binary ring R [0,1] 2,2 . Proposition 6.18. The polyadic ring R [a,b] m,n is a (m, n)-integral domain. Proof. It follows from the definitions (6.2)–(6.3), the condition a = 0, and commutativity and can- cellativity in Z.
  • 35. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 35 Lemma 6.19. There are no such congruence classes which can be described by polyadic (m, n)-field. Proof. Follows from Proposition 6.15 and Corollary 6.16. This statement for the limiting case [[1]]2 appeared in DUPLIJ AND WERNER [2015], while study- ing the ideal structure of the corresponding (3, 2)-ring. Proposition 6.20. In the limiting case a = 1 the polyadic ring R [1,b] b+1,2 can be embedded into a (b + 1, 2)-ary field. Proof. Because the polyadic ring R [1,b] b+1,2 of the congruence class [[1]]b is an (b + 1, 2)-integral domain by Proposition 6.18, we can construct in a standard way the correspondent (b + 1, 2)-quotient ring which is a (b + 1, 2)-ary field up to isomorphism, as was shown in CROMBEZ AND TIMM [1972]. By analogy, it can be called the field of polyadic rational numbers which have the form x = 1 + bk1 1 + bk2 , ki ∈ Z. (6.16) Indeed, they form a (b + 1, 2)-field, because each element has its inverse under multiplication (which is obvious) and additively “querable”, such that the equation for the querelement ¯x becomes νb+1 b x, x, . . . , x, ¯x = x which can be solved for any x, giving uniquely ¯x = − (b − 1) 1 + bk1 1 + bk2 . The introduced polyadic inner structure of the residue (congruence) classes allows us to extend various number theory problems by considering the polyadic (m, n)-integers Z(m,n) instead of Z. 7. EQUAL SUMS OF LIKE POWERS DIOPHANTINE EQUATION OVER POLYADIC INTEGERS First, recall the standard binary version of the equal sums of like powers Diophantine equation LANDER ET AL. [1967], EKL [1998]. Take the fixed non-negative integers p, q, l ∈ N0 , p ≤ q, and the positive integer unknowns ui, vj ∈ Z+, i = 1, . . . p + 1, j = 1, 1, . . . q + 1, then the Diophantine equation is p+1 i=1 ul+1 i = q+1 j=1 vl+1 j . (7.1) The trivial case, when ui = 0, vj = 0, for all i, j is not considered. We mark the solutions of (7.1) by the triple (l | p, q)r showing quantity of operations4 , where r (if it is used) is the order of the solution (ranked by the value of the sum) and the unknowns ui, vj are placed in ascending order ui ≤ ui+1, vj ≤ vj+1. Let us recall the Tarry-Escott problem (or multigrades problem) DORWART AND BROWN [1937]: to find the solutions to (7.1) for an equal number of summands on both sides of p = q and s equations simultaneously, such that l = 0, . . . , s. Known solutions exist for powers until s = 10, which are bounded such that s ≤ p (in our notations), see, also, NGUYEN [2016]. The solutions with highest powers s = p are the most interesting and called the ideal solutions BORWEIN [2002]. Theorem 7.1 (Frolov FROLOV [1889]). If the set of s Diophantine equations (7.1) with p = q for l = 0, . . . , s has a solution {ui, vi, i = 1, . . . p + 1}, then it has the solution {a + bui, a + bvi, i = 1, . . . p + 1}, where a, b ∈ Z are arbitrary and fixed. 4 In the binary case, the solutions of (7.1) are usually denoted by (l + 1 | p + 1, q + 1)r, which shows the number of summands on both sides and powers of elements LANDER ET AL. [1967]. But in the polyadic case (see below), the number of summands and powers do not coincide with l + 1, p + 1, q + 1, at all.
  • 36. 36 STEVEN DUPLIJ In the simplest case (1 | 0, 1), one term in l.h.s., one addition on the r.h.s. and one multiplication, the (coprime) positive numbers satisfying (7.1) are called a (primitive) Pythagorean triple. For the Fermat’s triple (l | 0, 1) with one addition on the r.h.s. and more than one multiplication l ≥ 2, there are no solutions of (7.1) , which is known as Fermat’s last theorem proved in WILES [1995]. There are many solutions known with more than one addition on both sides, where the highest number of multiplications till now is 31 (S. Chase, 2012). Before generalizing (7.1) for polyadic case we note the following. Remark 7.2. The notations in (7.1) are chosen in such a way that p and q are numbers of binary additions on both sides, while l is the number of binary multiplications in each term, which is natural for using polyadic powers DUPLIJ [2012]. 7.1. Polyadic analog of the Lander-Parkin-Selfridge conjecture. In LANDER ET AL. [1967], a generalization of Fermat’s last theorem was conjectured, that the solutions of (7.1) exist for small powers only, which can be formulated in terms of the numbers of operations as Conjecture 7.3 (Lander-Parkin-Selfridge LANDER ET AL. [1967]). There exist solutions of (7.1) in positive integers, if the number of multiplications is less than or equal than the total number of additions plus one 3 ≤ l ≤ lLSP = p + q + 1, (7.2) where p + q ≥ 2. Remark 7.4. If the equation (7.1) is considered over the binary ring of integers Z, such that ui, vj ∈ Z, it leads to a straightforward reformulation: for even powers it is obvious, but for odd powers all negative terms can be rearranged and placed on the other side. Let us consider the Diophantine equation (7.1) over polyadic integers Z(m,n) (i.e. over the polyadic (m, n)-ary ring RZ m,n) such that ui, vj ∈ RZ m,n. We use the “long products” µ (l) n and ν (l) m containing l operations, and also the “polyadic power” for an element x ∈ RZ m,n with respect to n-ary multiplica- tion DUPLIJ [2012] x l n = µ(l) n   l(n−1)+1 x, x, . . . , x   . (7.3) In the binary case, n = 2, the polyadic power coincides with (l + 1) power of an element x l 2 = xl+1 , which explains REMARK 7.2. In this notation the polyadic analog of the equal sums of like powers Diophantine equation has the form ν(p) m u l n 1 , u l n 2 , . . . , u l n p(m−1)+1 = ν(q) m v l n 1 , v l n 2 , . . . , v l n q(m−1)+1 , (7.4) where p and q are number of m-ary additions in l.h.s. and r.h.s. correspondingly. The solutions of (7.4) will be denoted by u1, u2, . . . , up(m−1)+1; v1, v2, . . . , vq(m−1)+1 . In the binary case m = 2, n = 2, (7.4) reduces to (7.1). Analogously, we mark the solutions of (7.4) by the polyadic triple (l | p, q)(m,n) r . Now the polyadic Pythagorean triple (1 | 0, 1)(m,n) , having one term on the l.h.s., one m-ary addition on the r.h.s. and one n-ary multiplication (elements are in the first polyadic power 1 n), becomes u 1 n 1 = νm v 1 n 1 , v 1 n 2 , . . . , v 1 n m . (7.5) Definition 7.5. The equation (7.5) solved by minimal u1, vi ∈ Z, i = 1, . . . , m can be named the polyadic Pythagorean theorem.
  • 37. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 37 The polyadic Fermat’s triple (l | 0, 1)(m,n) has one term in l.h.s., one m-ary addition in r.h.s. and l (n-ary) multiplications u l n 1 = νm v l n 1 , v l n 2 , . . . , v l n m . (7.6) One may be interested in whether the polyadic analog of Fermat’s last theorem is valid, and if not, in which cases the analogy with the binary case can be sustained. Conjecture 7.6 (Polyadic analog of Fermat’s Last Theorem). The polyadic Fermat’s triple (7.6) has no solutions over the polyadic (m, n)-ary ring RZ m,n, if l ≥ 2, i.e. there are more than one n-ary multiplications. Its straightforward generalization leads to the polyadic version of the Lander-Parkin-Selfridge con- jecture, as Conjecture 7.7 (Polyadic Lander-Parkin-Selfridge conjecture). There exist solutions of the polyadic analog of the equal sums of like powers Diophantine equation (7.4) in integers, if the number of n-ary multiplications is less than or equal than the total number of m-ary additions plus one 3 ≤ l ≤ lpLP S = p + q + 1. (7.7) Below we will see a counterexample to both of the above conjectures. Example 7.8. Let us consider the (3, 2)-ring RZ 3,2 = Z | ν3, µ2 , where ν3 [x, y, z] = x + y + z + 2, (7.8) µ2 [x, y] = xy + x + y. (7.9) Note that this exotic polyadic ring is commutative and cancellative, having unit 0, no multiplicative inverses, and for any x ∈ RZ 3,2 its additive querelement ˜x = −x − 2, therefore Z | ν3 is a ternary group (as it should be). The polyadic power of any element is x l 2 = (x + 1)l+1 − 1. (7.10) 1) For RZ 3,2 the polyadic Pythagorean triple (1 | 0, 1)(3,2) in (7.5) now is u 1 2 = ν3 x 1 2 , y 1 2 , z 1 2 , (7.11) which, using (7.3), (7.9) and (7.10), becomes the (shifted) Pythagorean quadruple SPIRA [1962] (u + 1)2 = (x + 1)2 + (y + 1)2 + (z + 1)2 , (7.12) and it has infinite number of solutions, among which two minimal ones {u = 2; x = 0, y = z = 1} and {u = 14; x = 1, y = 9, z = 10} give 32 = 12 +22 +22 and 152 = 22 +102 +112 , correspondingly. 2) For this (3, 2)-ring RZ 3,2 the polyadic Fermat’s triple (l | 0, 1)(3,2) becomes (u + 1)l+1 = (x + 1)l+1 + (y + 1)l+1 + (z + 1)l+1 . (7.13) If the polyadic analog of Fermat’s last theorem 7.6 holds, then there are no solutions to (7.13) for more than one n-ary multiplication l ≥ 2. But this is the particular case, p = 0, q = 2, of the binary Lander-Parkin-Selfridge Conjecture 7.3 which now takes the form: the solutions to (7.13) exist, if l ≤ 3. Thus, as a counterexample to the polyadic analog of Fermat’s last theorem, we have two possible solutions with numbers of multiplications: l = 2, 3. In the case of l = 2 there exist two solutions: one well-known solution {u = 5; x = 2, y = 3, z = 4} giving 63 = 33 + 43 + 53 and another one giving 7093 = 1933 +4613 +6313 (J.-C. Meyrignac, 2000), while for l = 3 there exist an infinite number of solutions, and one of them (minimal) gives 4224814 = 958004 +2175194 +4145604 ELKIES [1988].
  • 38. 38 STEVEN DUPLIJ 3) The general polyadic triple (l | p, q)(3,2) , using (7.4), can be presented in the standard binary form (as (7.1)) 2p+1 i=1 (ui + 1)l+1 = 2q+1 j=1 (vi + 1)l+1 , ui, vj ∈ Z. (7.14) Let us apply the polyadic Lander-Parkin-Selfridge Conjecture 7.7 for this case: the solutions to (7.14) exist, if 3 ≤ l ≤ lpLSP = p + q + 1. But the binary Lander-Parkin-Selfridge Con- jecture 7.3, applied directly, gives 3 ≤ l ≤ lLSP = 2p + 2q + 1. So we should have coun- terexamples to the polyadic Lander-Parkin-Selfridge conjecture, when lpLSP < l ≤ lLSP . For in- stance, for p = q = 1, we have lpLSP = 3, while the (minimal) counterexample with l = 5 is {u1 = 3, u2 = 18, u3 = 21, v1 = 9, v2 = 14, v3 = 22} giving 36 + 196 + 226 = 106 + 156 + 236 SUBBA RAO [1934]. As it can be observed from Example 7.8, the arity shape of the polyadic ring RZ m,n is crucial in constructing polyadic analogs of the equal sums of like powers conjectures. We can make some general estimations assuming a special (more or less natural) form of its operations over integers. Definition 7.9. We call RZ m,n the standard polyadic ring, if the “leading terms” of its m-ary addition and n-ary multiplication are νm m x, x, . . . , x ∼ mx, (7.15) µn n x, x, . . . , x ∼ xn , x ∈ Z. (7.16) The polyadic ring RZ 3,2 from Example 7.8, as well as the congruence class polyadic ring R [a,b] m,n (6.9) are both standard. Using (7.3), we obtain approximate behavior of the polyadic power in the standard polyadic ring x l n ∼ xl(n−1)+1 , x ∈ Z, l ∈ N, n ≥ 2. (7.17) So increasing the arity of multiplication leads to higher powers, while increasing arity of addition gives more terms in sums. Thus, the estimation for the polyadic analog of the equal sums of like powers Diophantine equation (7.4) becomes (p (m − 1) + 1) xl(n−1)+1 ∼ (q (m − 1) + 1) xl(n−1)+1 , x ∈ Z. (7.18) Now we can apply the binary Lander-Parkin-Selfridge Conjecture 7.3 in the form: the solutions to (7.18) can exist if 3 ≤ l ≤ lLP S, where lLP S is an integer solution of (n − 1) lLP S = (p + q) (m − 1) + 1. (7.19) On the other hand the polyadic Lander-Parkin-Selfridge Conjecture 7.7 gives: the solutions to (7.18) can exist if 3 ≤ l ≤ lpLP S = p + q + 1. Note that (p + q) ≥ 2 now. An interesting question arises: which arities give the same limit, that is, when lpLP S = lLP S? Proposition 7.10. For any fixed number of additions in both sizes of the polyadic analog of the equal sums of like powers Diophantine equation (7.4) p + q ≥ 2, there exist limiting arities m0 and n0 (excluding the trivial binary case m0 = n0 = 2), for which the binary and polyadic Lander-Parkin- Selfridge conjectures coincide lpLP S = lLP S, such that m0 = 3 + p + q + (p + q + 1) k, (7.20) n0 = 2 + p + q + (p + q) k, k ∈ N0 . (7.21)
  • 39. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 39 Proof. To equate lLP S = lpLP S = p + q + 1 we use (7.19) and solve in integers the equation (n0 − 1) (p + q + 1) = (p + q) (m0 − 1) + 1. (7.22) In the trivial case, m0 = n0 = 2, this is an identity, while the other solutions can be found from n0 (p + q + 1) = (p + q) m0 + 2, which gives (7.20)–(7.21). Corollary 7.11. In the limiting case lpLP S = lLP S the arity of multiplication always exceeds the arity of addition m0 − n0 = k + 1, k ∈ N0 , (7.23) and they start from m0 ≥ 5, n0 ≥ 4. The first allowed arities m0 and n0 are presented in TABLE 4. Their meaning is the following. Corollary 7.12. For the polyadic analog of the equal sums of like powers equation over the standard polyadic ring RZ m,n (with fixed p + q ≥ 2) the polyadic Lander-Parkin-Selfridge conjecture becomes weaker than the binary one lpLP S ≥ lLP S, if: 1) the arity of multiplication exceeds its limiting value n0 with fixed arity of the addition; 2) the arity of addition is lower, than its limiting value m0 with the fixed arity of multiplication. TABLE 4. The limiting arities m0 and n0, which give lpLP S = lLP S in (7.18). p + q = 2 p + q = 3 p + q = 4 m0 n0 m0 n0 m0 n0 5 4 6 5 7 6 8 6 10 8 12 10 11 8 14 11 17 14 14 10 18 14 22 18 Example 7.13. Consider the standard polyadic ring RZ m,n and fix the arity of addition m0 = 12, then take in (7.18) the total number of additions p + q = 4 (the last column in TABLE 4). We observe that the arity of multiplication n = 16, which exceeds the limiting arity n0 = 10 (corresponding to m0). Thus, we obtain lpLP S = 5 and lLP S = 3 by solving (7.19) in integers, and therefore the polyadic Lander-Parkin-Selfridge conjecture becomes now weaker than the binary one, and we do not obtain counterexamples to it, as in Example 7.8 (where the situation was opposite lpLP S = 3 and lLP S = 5, and they cannot be equal). A concrete example of the standard polyadic ring (Definition 7.9) is the polyadic ring of the fixed congruence class R [a,b] m,n considered in SECTION 6, because its operations (6.2)–(6.3) have the same straightforward behavior (7.15)–(7.16). Let us formulate the polyadic analog of the equal sums of like powers Diophantine equation (7.4) over R [a,b] m,n in terms of operations in Z. Using (6.2)–(6.3) and (7.17) for (7.4) we obtain p(m−1)+1 i=1 (a + bki)l(n−1)+1 = q(m−1)+1 j=1 (a + bkj)l(n−1)+1 , a, b, ki ∈ Z. (7.24) It is seen that the leading power behavior of both sides in (7.24) coincides with the general es- timation (7.18). But now the arity shape (m, n) is fixed by (6.10)–(6.11) and given in TABLE 3. Nevertheless, we can consider for (7.24) the polyadic analog of Fermat’s last theorem 7.6, the Lander- Parkin-Selfridge Conjecture 7.3 (solutions exist for l ≤ lLP S) and its polyadic version (Conjecture 7.7, solutions exist for l ≤ lpLP S), as in the estimations above. Let us consider some examples of solutions to (7.24).
  • 40. 40 STEVEN DUPLIJ Example 7.14. Let [[2]]3 be the congruence class, which is described by (4, 3)-ring R [2,3] 4,3 (see TABLE 3), and we consider the polyadic Fermat’s triple (l | 0, 5)(4,3) (7.6). Now the powers are lLP S = 8, lpLP S = 6, and for instance, if l = 2, we have solutions, because l < lpLP S < lLP S, and one of them is 145 = 4 · (−1)5 + 7 · 55 + 85 + 2 · 115 . (7.25) 7.2. Frolov’s Theorem and the Tarry-Escott problem. A special set of solutions to the polyadic Lander-Parkin-Selfridge Conjecture 7.7 can be generated, if we put p = q in (7.24), which we call equal-summand solutions5 , by exploiting the Tarry-Escott problem approach DORWART AND BROWN [1937] and Frolov’s Theorem 7.1. Theorem 7.15. If the set of integers ki ∈ Z solves the Tarry-Escott problem p(m−1)+1 i=1 kr i = p(m−1)+1 j=1 kr j , r = 1, . . . , s = l (n − 1) + 1, (7.26) then the polyadic equal sums of like powers equation with equal summands (7.4) has a solution over the polyadic (m, n)-ring R [a,b] m,n having the arity shape given by the following relations: 1) Inequality l (n − 1) + 1 ≤ p (m − 1) ; (7.27) 2) Equality p (m − 1) = 2l(n−1)+1 . (7.28) Proof. Using Frolov’s Theorem 7.1 applied to (7.26) we state that p(m−1)+1 i=1 (a + bki)r = p(m−1)+1 j=1 (a + bkj)r , r = 1, . . . , s = l (n − 1) + 1, (7.29) for any fixed integers a, b ∈ Z. This means, that (7.29), with ki (satisfying (7.26)) corresponds to a solution to the polyadic equal sums of like powers equation (7.4) for any congruence class [[a]]b. Nevertheless, the values a and b are fixed by the restrictions on the arity shape and the relations (6.4) and (6.8). 1) It is known that the Tarry-Escott problem can have a solution only when the powers are strongly less than the number of summands BORWEIN [2002], DORWART AND BROWN [1937], that is (l (n − 1) + 1) + 1 ≤ p (m − 1) + 1, which gives (7.27). 2) A special kind of solutions, when number of summands is equal to 2 into the number of powers, was found using the Thue–Morse sequence ALLOUCHE AND SHALLIT [1999], which always satisfies the bound (7.27), and in our notation it is (7.28). In both cases the relations (7.27) and (7.28) should be solved in positive integers and with m ≥ 2 and n ≥ 2, which can lead to non-unique solutions. Let us consider some examples which give solutions to the polyadic equal sums of like powers equation (7.4) with p = q over the polyadic (m, n)-ring R [a,b] m,n of the fixed congruence class [[a]]b. Example 7.16. 1) One of the first ideal (non-symmetric) solutions to the Tarry-Escott problem has 6 summands and 5 powers (A. Golden, 1944) 0r + 19r + 25r + 57r + 62r + 86r = 2r + 11r + 40r + 42r + 69r + 85r , r = 1, . . . , 5. (7.30) 5 The term “symmetric solution” is already taken and widely used BORWEIN [2002].
  • 41. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 41 We compare with (7.26) and obtain p (m − 1) = 5, (7.31) l (n − 1) = 4. (7.32) After ignoring binary arities we get m = 6, p = 1 and n = 3, l = 2. From Theorem 6.8 and TABLE 3 we observe the minimal choice a = 4 and b = 5. It follows from Frolov’s Theorem 7.1, that all equations in (7.30) have symmetry ki → a + bki = 4 + 5ki. Thus, we obtain the solution of the polyadic equal sums of like powers equation (7.4) for the fixed congruence class [[4]]5 in the form 45 + 995 + 1295 + 2895 + 3145 + 4345 = 145 + 595 + 2045 + 2145 + 3495 + 4295 . (7.33) It is seen from TABLE 3 that the arity shape (m = 6, n = 3) corresponds, e.g., to the congruence class [[4]]10 as well. Using Frolov’s theorem, we substitute in (7.30) ki → 4 + 10ki to obtain the solution in the congruence class [[4]]10 45 + 1945 + 2545 + 5745 + 6245 + 8645 = 245 + 1145 + 4045 + 4245 + 6945 + 8545 . (7.34) 2) To obtain the special kind of solutions to the Tarry-Escott problem we start with the known one with 8 summands and 3 powers (see, e.g., LEHMER [1947]) 0r +3r +5r +6r +9r +10r +12r +15r = 1r +2r +4r +7r +8r +11r +13r +14r , r = 1, 2, 3. (7.35) So we have the concrete solution to the system (7.26) with the condition (7.28) which now takes the form 8 = 23 , and therefore p (m − 1) = 7, (7.36) l (n − 1) = 2. (7.37) Excluding the trivial case containing binary arities, we have m = 8, p = 1 and n = 3, l = 1. It follows from Theorem 6.8 and TABLE 3, that a = 6 and b = 7, and so the polyadic ring is R [6,7] 8,3 . Using Frolov’s Theorem 7.1, we can substitute entries in (7.35) as ki → a + bki = 6 + 7ki in the equation with highest power r = 3 (which is relevant to our task) and obtain the solution of (7.4) for [[6]]7 as follows 63 +273 +413 +483 +693 +763 +903 +1113 = 133 +203 +343 +553 +623 +833 +973 +1043 . (7.38) We conclude that consideration of the Tarry-Escott problem and Frolov’s theorem over polyadic rings gives the possibility of obtaining many nontrivial solutions to the polyadic equal sums of like powers equation for fixed congruence classes. Acknowledgments. The author would like to express his deep gratitude and sincere thankfulness to Joachim Cuntz, Christopher Deninger, Grigorij Kurinnoj, Mike Hewitt, Jim Stasheff, Alexander Voronov, and Wend Werner for discussions, and to Dara Shayda for Mathematica programming help.
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  • 43. ARITY SHAPE OF POLYADIC ALGEBRAIC STRUCTURES 43 LEHMER, D. H. (1947). The Tarry-Escott problem. Scripta Math. 13, 37–41. LISTER, W. G. (1971). Ternary rings. Trans. Amer. Math. Soc. 154, 37–55. MICHALSKI, J. (1988). On J-derived polyadic groups. Mathematica (Cluj) 30 (53), 149–155. MIGNOT, F. (1976). Contrˆole dans les in´equations variationelles elliptiques. J. Funct. Anal. 22, 130–185. MONK, J. D. AND F. SIOSON (1966). m-semigroups, semigroups and function representations. Fundam. Math. 59, 233–241. NGUYEN, H. D. (2016). A new proof of the Prouhet-Tarry-Escott problem. Integers 16, A01, 1–9. POP, A. AND M. S. POP (2002). Some embeddings theorems for (n, 2)-rings. Bul. S¸tiint¸. Univ. Baia Mare, Ser. B, Fasc. Mat.-Inform. 18 (2), 311–316. RUSAKOV, S. A. (1998). Some Applications of n-ary Group Theory. Minsk: Belaruskaya navuka. SPIRA, R. (1962). The Diophantine equation x2 + y2 + z2 = m2. Am. Math. Mon. 69, 360–365. SUBBA RAO, K. (1934). On sums of sixth powers. J. London Math. Soc. s1-9 (3), 172–173. WILES, A. (1995). Modular elliptic curves and fermat’s last theorem. Ann. Math. 141 (3), 443–551. ZUPNIK, D. (1967). Polyadic semigroups. Publ. Math. (Debrecen) 14, 273–279. LIST OF TABLES 1 “Quantization” of arity ℓ-shapes 11 2 The arity signature and arity shape of polyadic algebraic structures. 21 3 The polyadic ring R Z(a,b) m,n of the fixed residue class [[a]]b: arity shape. 34 4 The limiting arities m0 and n0, which give lpLP S = lLP S in (7.18). 39 MATHEMATISCHES INSTITUTE, UNIVERSIT ¨AT M ¨UNSTER EINSTEINSTR. 62, D-48149 M ¨UNSTER, DEUTSCHLAND E-mail address: duplijs@math.uni-muenster.de, sduplij@gmail.com URL: http://homepages.spa.umn.edu/~duplij