S. Duplij - Polyadic systems, representations and quantum groups

arXiv:1308.4060v2[math.RT]8Jun2014
POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS
STEVEN DUPLIJ
ABSTRACT. Polyadic systems and their representations are reviewed and a classification of general
polyadic systems is presented. A new multiplace generalization of associativity preserving homomor-
phisms, a ’heteromorphism’ which connects polyadic systems having unequal arities, is introduced
via an explicit formula, together with related definitions for multiplace representations and multiac-
tions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary
algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary gen-
eralizations of quantum groups and the Yang-Baxter equation are presented.
CONTENTS
1. INTRODUCTION 2
2. PRELIMINARIES 3
3. SPECIAL ELEMENTS AND PROPERTIES OF POLYADIC SYSTEMS 6
4. HOMOMORPHISMS OF POLYADIC SYSTEMS 11
5. MULTIPLACE MAPPINGS OF POLYADIC SYSTEMS AND HETEROMORPHISMS 13
6. ASSOCIATIVITY QUIVERS AND HETEROMORPHISMS 18
7. MULTIPLACE REPRESENTATIONS OF POLYADIC SYSTEMS 24
8. MULTIACTIONS AND G-SPACES 29
9. REGULAR MULTIACTIONS 31
10. MULTIPLACE REPRESENTATIONS OF TERNARY GROUPS 33
11. MATRIX REPRESENTATIONS OF TERNARY GROUPS 37
12. TERNARY ALGEBRAS AND HOPF ALGEBRAS 40
13. TERNARY QUANTUM GROUPS 44
14. CONCLUSIONS 47
REFERENCES 47
Date: 20 May 2012.
2010 Mathematics Subject Classification. 16T05, 16T25, 17A42, 20N15, 20F29, 20G05, 20G42, 57T05.
Published: Journal of Kharkov National University, (ser. Nuclei, Particles and Fields), Vol. 1017, 3(55) (2012) 28-59.
1

2 STEVEN DUPLIJ
1. INTRODUCTION
One of the most promising directions in generalizing physical theories is the consideration of higher
arity algebras KERNER [2000], in other words ternary and n-ary algebras, in which the binary com-
position law is substituted by a ternary or n-ary one DE AZCARRAGA AND IZQUIERDO [2010].
Firstly, ternary algebraic operations (with the arity n = 3) were introduced already in the XIX-
th century by A. Cayley in 1845 and later by J. J. Silvester in 1883. The notion of an n-ary group
was introduced in 1928 by D ÖRNTE [1929] (inspired by E. Nöther) and is a natural generalization
of the notion of a group. Even before this, in 1924, a particular case, that is, the ternary group
of idempotents, was used in PR ÜFER [1924] to study infinite abelian groups. The important coset
theorem of Post explained the connection between n-ary groups and their covering binary groups
POST [1940]. The next step in study of n-ary groups was the Gluskin-Hosszú theorem HOSSZ Ú
[1963], GLUSKIN [1965]. Another definition of n-ary groups can be given as a universal algebra with
additional laws DUDEK ET AL. [1977] or identities containing special elements RUSAKOV [1979].
The representation theory of (binary) groups WEYL [1946], FULTON AND HARRIS [1991] plays
an important role in their physical applications CORNWELL [1997]. It is initially based on a matrix
realization of the group elements with the abstract group action realized as the usual matrix multi-
plication CURTIS AND REINER [1962], COLLINS [1990]. The cubic and n-ary generalizations of
matrices and determinants were made in KAPRANOV ET AL. [1994], SOKOLOV [1972], and their
physical application appeared in KAWAMURA [2003], RAUSCH DE TRAUBENBERG [2008]. In gen-
eral, particular questions of n-ary group representations were considered, and matrix representations
derived, by the author BOROWIEC ET AL. [2006], and some general theorems connecting representa-
tions of binary and n-ary groups were presented in DUDEK AND SHAHRYARI [2012]. The intention
here is to generalize the above constructions of n-ary group representations to more complicated and
nontrivial cases.
In physics, the most applicable structures are the nonassociative Grassmann, Clifford and Lie al-
gebras L ÕHMUS ET AL. [1994], LOUNESTO AND ABLAMOWICZ [2004], GEORGI [1999], and so
their higher arity generalizations play the key role in further applications. Indeed, the ternary ana-
log of Clifford algebra was considered in ABRAMOV [1995], and the ternary analog of Grassmann
algebra ABRAMOV [1996] was exploited to construct various ternary extensions of supersymmetry
ABRAMOV ET AL. [1997].
The construction of realistic physical models is based on Lie algebras, such that the fields take their
values in a concrete binary Lie algebra GEORGI [1999]. In the higher arity studies, the standard Lie
bracket is replaced by a linear n-ary bracket, and the algebraic structure of the corresponding model
is defined by the additional characteristic identity for this generalized bracket, corresponding to the
Jacobi identity DE AZCARRAGA AND IZQUIERDO [2010]. There are two possibilities to construct
the generalized Jacobi identity: 1) The Lie bracket is a derivation by itself; 2) A double Lie bracket
vanishes, when antisymmetrized with respect to its entries. The first case leads to the so called
Filippov algebras FILIPPOV [1985] (or n-Lie algebra) and second case corresponds to generalized
Lie algebras MICHOR AND VINOGRADOV [1996] (or higher order Lie algebras).
The infinite-dimensional version of n-Lie algebras are the Nambu algebras NAMBU [1973],
TAKHTAJAN [1994], and their n-bracket is given by the Jacobian determinant of n functions,
the Nambu bracket, which in fact satisfies the Filippov identity FILIPPOV [1985]. Recently,
the ternary Filippov algebras were successfully applied to a three-dimensional superconformal
gauge theory describing the effective worldvolume theory of coincident M2-branes of M-theory
BAGGER AND LAMBERT [2008a,b], GUSTAVSSON [2009]. The infinite-dimensional Nambu

POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 3
bracket realization HO ET AL. [2008] gave the possibility to describe a condensate of nearly co-
incident M2-branes LOW [2010].
From another side, Hopf algebras ABE [1980], SWEEDLER [1969], MONTGOMERY [1993]
play a fundamental role in quantum group theory KASSEL [1995], SHNIDER AND STERNBERG
[1993]. Previously, their Von Neumann generalization was introduced in DUPLIJ AND LI [2001],
DUPLIJ AND SINEL’SHCHIKOV [2009], LI AND DUPLIJ [2002], their actions on the quantum plane
were classified in DUPLIJ AND SINEL’SHCHIKOV [2010], and ternary Hopf algebras were defined
and studied in DUPLIJ [2001], BOROWIEC ET AL. [2001].
The goal of this paper is to give a comprehensive review of polyadic systems and their representa-
tions. First, we classify general polyadic systems and introduce n-ary semigroups and groups. Then
we consider their homomorphisms and multiplace generalizations, paying attention to their associa-
tivity. We define multiplace representations and multiactions, and give examples of matrix represen-
tations for some ternary groups. We define and investigate ternary algebras and Hopf algebras, study
their properties and give some examples. At the end we consider some ternary generalizations of
quantum groups and the Yang-Baxter equation.
2. PRELIMINARIES
Let G be a non-empty set (underlying set, universe, carrier), its elements we denote by lower-
case Latin letters gi ∈ G. The n-tuple (or polyad) g1, . . . , gn of elements from G is denoted by
(g1, . . . , gn). The Cartesian product1
n
G × . . . × G = G×n
consists of all n-tuples (g1, . . . , gn), such
that gi ∈ G, i = 1, . . . , n. For all equal elements g ∈ G, we denote n-tuple (polyad) by power (gn
).
If the number of elements in the n-tuple is clear from the context or is not important, we denote it
with one bold letter (g), in other cases we use the power in brackets g(n)
. We now introduce two
important constructions on sets.
Definition 2-1. The i-projection of the Cartesian product G×n
on its i-th “axis” is the map Pr
(n)
i :
G×n
→ G such that (g1, . . . gi, . . . , gn) −→ gi.
Definition 2-2. The i-diagonal Diagn : G → G×n
sends one element to the equal element n-tuple
g −→ (gn
).
The one-point set {•} can be treated as a unit for the Cartesian product, since there are bijections
between G and G × {•}×n
, where G can be on any place. On the Cartesian product G×n
one can
define a polyadic (n-ary, n-adic, if it is necessary to specify n, its arity or rank) operation µn :
G×n
→ G. For operations we use small Greek letters and place arguments in square brackets µn [g].
The operations with n = 1, 2, 3 are called unary, binary and ternary. The case n = 0 is special
and corresponds to fixing a distinguished element of G, a “constant” c ∈ G, and it is called a 0-ary
operation µ
(c)
0 , which maps the one-point set {•} to G, such that µ
(c)
0 : {•} → G, and formally has
the value µ
(c)
0 [{•}] = c ∈ G. The 0-ary operation “kills” arity, which can be seen from the following
BERGMAN [1995]: the composition of n-ary and m-ary operations µn ◦ µm gives (n + m − 1)-ary
operation by
µn+m−1 [g, h] = µn [g, µm [h]] . (2.1)
Then, if to compose µn with the 0-ary operation µ
(c)
0 , we obtain
µ
(c)
n−1 [g] = µn [g, c] , (2.2)
1We place the sign for the Cartesian product (×) into the power, because the same abbreviation will also be used below
for other types of product.

4 STEVEN DUPLIJ
because g is a polyad of length (n − 1). So, it is necessary to make a clear distinction between the
0-ary operation µ
(c)
0 and its value c in G, as will be seen and will become important below.
Definition 2-3. A polyadic system G is a set G which is closed under polyadic operations.
We will write G = set|operations or G = set|operations|relations , where “relations” are
some additional properties of operations (e.g., associativity conditions for semigroups or cancella-
tion properties). In such a definition it is not necessary to list the images of 0-ary operations (e.g.
the unit or zero in groups), as is done in various other definitions. Here, we mostly consider con-
crete polyadic systems with one “chief” (fundamental) n-ary operation µn, which is called polyadic
multiplication (or n-ary multiplication).
Definition 2-4. A n-ary system Gn = G | µn is a set G closed under one n-ary operation µn
(without any other additional structure).
Note that a set with one closed binary operation without any other relations was called a groupoid
by Hausmann and Ore HAUSMANN AND ORE [1937] (see, also CLIFFORD AND PRESTON [1961]).
However, nowadays the term “groupoid” is widely used in category theory and homotopy theory for
a different construction with binary multiplication, the so-called Brandt groupoid BRANDT [1927]
(see, also, BRUCK [1966]). Alternatively, and much later on, Bourbaki BOURBAKI [1998] intro-
duced the term “magma” for binary systems. Then, the above terms were extended to the case of
one fundamental n-ary operation as well. Nevertheless, we will use some neutral notations “polyadic
system” and “n-ary system” (when arity n is fixed/known/important), which adequately indicates all
of their main properties.
Let us consider the changing arity problem:
Definition 2-5. For a given n-ary system G | µn to construct another polyadic system G | µ′
n′
over the same set G, which has multiplication with a different arity n′
.
The formulas (2.1) and (2.2) give us the simplest examples of how to change the arity of a polyadic
system. In general, there are 3 ways:
(1) Iterating. Using composition of the operation µn with itself, one can increase the arity from n
to n′
iter (as in (2.1)) without changing the signature of the system. We denote the number of
iterating multiplications by ℓµ, and use the bold Greek letters µ
ℓµ
n for the resulting composition
of n-ary multiplications, such that
µ′
n′ = µℓµ
n
def
=
ℓµ
µn ◦ µn ◦ . . . µn × id×(n−1)
. . . × id×(n−1)
, (2.3)
where
n′
= niter = ℓµ (n − 1) + 1, (2.4)
which gives the length of a polyad (g) in the notation µ
ℓµ
n [g]. Without assuming associativity
there many variants for placing µn’s among id’s in the r.h.s. of (2.3). The operation µ
ℓµ
n is
named a long product D ÖRNTE [1929] or derived DUDEK [2007].

(2) Reducing (Collapsing). Using nc distinguished elements or constants (or nc additional 0-ary
operations µ
(ci)
0 , i = 1, . . . nc), one can decrease arity from n to n′
red (as in (2.2)), such that2
µ′
n′ = µ
(c1...cnc )
n′
def
= µn ◦


nc
µ
(c1)
0 × . . . × µ
(cnc )
0 × id×(n−nc)

 , (2.5)
where
n′
= nred = n − nc, (2.6)
and the 0-ary operations µ
(ci)
0 can be on any places.
(3) Mixing. Changing (increasing or decreasing) arity may be done by combining iterating and
reducing (maybe with additional operations of different arity). If we do not use additional
operations, the ﬁnal arity can be presented in a general form using (2.4) and (2.6). It will
depend on the order of iterating and reducing, and so we have two subcases:
(a) Iterating→Reducing. We have
n′
= niter→red = ℓµ (n − 1) − nc + 1. (2.7)
The maximal number of constants (when n′
iter→red = 2) is equal to
nmax
c = ℓµ (n − 1) − 1 (2.8)
and can be increased by increasing the number of multiplications ℓµ.
(b) Reducing→Iterating. We obtain
n′
= nred→iter = ℓµ (n − 1 − nc) + 1. (2.9)
Now the maximal number of constants is
nmax
c = n − 2 (2.10)
and this is achieved only when ℓµ = 1.
To give examples of the third (mixed) case we put n = 4, ℓµ = 3, nc = 2 for both subcases of
opposite ordering:
(1) Iterating→Reducing. We can put
µ
(c1,c2)′
8 g(8)
= µ4 [g1, g2, g3, µ4 [g4, g5, g6, µ4 [g7, g8, c1, c2]]] , (2.11)
which corresponds to the following commutative diagram
G×8 ǫ
✲ G×8
× {•}2 id×6
×µ4
✲ G×7 id×3
×µ4
✲ G×4
PPPPPPPPPPPPPPPPPPP
µ
(c1,c2)′
8
q
G
µ4
❄
(2.12)
(2) Reducing→Iterating. We can have
µ
(c1,c2)′
4 g(4)
= µ4 [g1, c1, c2, µ4 [g2, c1, c2, µ4 [g3, c1, c2, g4]]] , (2.13)
2In DUDEK AND MICHALSKI [1984] µ
(c1...cnc )
n is named a retract (which term is already busy and widely used in
category theory for another construction).

6 STEVEN DUPLIJ
such that the diagram
G×4 ǫ
✲ G × {•}2 ×3
× G
id×6
×µ4
✲ G×7 id×3
×µ4
✲ G×4
❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳
µ
(c1,c2)′
4
③
G
µ4
❄
(2.14)
is commutative.
It is important to find conditions where iterating and reducing compensate each other, i.e. they
do not change arity overall. Indeed, let the number of the iterating multiplications ℓµ be fixed, then
we can find such a number of reducing constants n
(0)
c , such that the final arity will coincide with the
initial arity n. The result will depend on the order of operations. There are two cases:
(1) Iterating→Reducing. For the number of reducing constants n
(0)
c we obtain from (2.4) and
(2.6)
n(0)
c = (n − 1) (ℓµ − 1) , (2.15)
such that there is no restriction on ℓµ.
(2) Reducing→Iterating. For n
(0)
c we get
n(0)
c =
(n − 1) (ℓµ − 1)
ℓµ
, (2.16)
and now ℓµ ≤ n − 1. The requirement that n
(0)
c should be an integer gives two further
possibilities
n(0)
c =
n − 1
2
, ℓµ = 2,
n − 2, ℓµ = n − 1.
(2.17)
The above relations can be useful in the study of various n-ary multiplication structures and their
presentation in special form is needed in concrete problems.
3. SPECIAL ELEMENTS AND PROPERTIES OF POLYADIC SYSTEMS
Let us recall the definitions of some standard algebraic systems and their special elements, which
will be considered in this paper, using our notation.
Definition 3-1. A zero of a polyadic system is a distinguished element z (and the corresponding
0-ary operation µ
(z)
0 ) such that for any (n − 1)-tuple (polyad) g ∈ G×(n−1)
we have
µn [g, z] = z, (3.1)
where z can be on any place in the l.h.s. of (3.1).
There is only one zero (if its place is not fixed) which can be possible in a polyadic system. As in
the binary case, an analog of positive powers of an element POST [1940] should coincide with the
number of multiplications ℓµ in the iterating (2.3).
Definition 3-2. A (positive) polyadic power of an element is
g ℓµ
= µℓµ
n gℓµ(n−1)+1
. (3.2)

Definition 3-3. An element of a polyadic system g is called ℓµ-nilpotent (or simply nilpotent for
ℓµ = 1), if there exist such ℓµ that
g ℓµ
= z. (3.3)
Definition 3-4. A polyadic system with zero z is called ℓµ-nilpotent, if there exists ℓµ such that for
any (ℓµ (n − 1) + 1)-tuple (polyad) g we have
µℓµ
n [g] = z. (3.4)
Therefore, the index of nilpotency (number of elements whose product is zero) of an ℓµ-
nilpotent n-ary system is (ℓµ (n − 1) + 1), while its polyadic power is ℓµ .
Definition 3-5. A polyadic (n-ary) identity (or neutral element) of a polyadic system is a dis-
tinguished element e (and the corresponding 0-ary operation µ
(e)
0 ) such that for any element g ∈ G
we have
µn g, en−1
= g, (3.5)
where g can be on any place in the l.h.s. of (3.5).
In binary groups the identity is the only neutral element, while in polyadic systems, there exist
neutral polyads n consisting of elements of G satisfying
µn [g, n] = g, (3.6)
where g can be also on any place. The neutral polyads are not determined uniquely. It follows from
(3.5) that the sequence of polyadic identities en−1
is a neutral polyad.
Definition 3-6. An element of a polyadic system g is called ℓµ-idempotent (or simply idempotent
for ℓµ = 1), if there exist such ℓµ that
g ℓµ
= g. (3.7)
Both zero and the identity are ℓµ-idempotents with arbitrary ℓµ. We define (total)
associativity as the invariance of the composition of two n-ary multiplications
µ2
n [g, h, u] = µn [g, µn [h] , u] = invariant (3.8)
under placement of the internal multiplication in r.h.s. with a fixed order of elements in the whole
polyad of (2n − 1) elements t(2n−1)
= (g, h, u). Informally, “internal brackets/multiplication can be
moved on any place”, which gives n relations
µn ◦ µn × id×(n−1)
= . . . = µn ◦ id×(n−1)
×µn . (3.9)
There are many other particular kinds of associativity which were introduced in THURSTON [1949]
and studied in BELOUSOV [1972], SOKHATSKY [1997]. Here we will confine ourselves the most
general, total associativity (3.8). In this case, the iteration does not depend on the placement of
internal multiplications in the r.h.s of (2.3).
Definition 3-7. A polyadic semigroup (n-ary semigroup) is a n-ary system in which the opera-
tion is associative, or Gsemigrp
n = G | µn | associativity .
In a polyadic system with zero (3.1) one can have trivial associativity, when all n terms are
(3.8) are equal to zero, i.e.
µ2
n [g] = z (3.10)
for any (2n − 1)-tuple g. Therefore, we state that

8 STEVEN DUPLIJ
Assertion 3-8. Any 2-nilpotent n-ary system (having index of nilpotency (2n − 1)) is a polyadic
semigroup.
In the case of changing arity one should use in (3.10) not the changed final arity n′
, but the “real”
arity which is n for the reducing case and ℓµ (n − 1)+1 for all other cases. Let us give some examples.
Example 3-9. In the mixed (interacting-reducing) case with n = 2, ℓµ = 3, nc = 1, we have a ternary
system G | µ3 iterated from a binary system G | µ2, µ
(c)
0 with one distinguished element c (or an
additional 0-ary operation)3
µ
(c)
3 [g, h, u] = (g · (h · (u · c))) , (3.11)
where for binary multiplication we denote g · h = µ2 [g, h]. Thus, if the ternary system G | µ
(c)
3
is nilpotent of index 7 (see 3-4), then it is a ternary semigroup (because µ
(c)
3 is trivially associative)
independently of the associativity of µ2 (see, e.g. BOROWIEC ET AL. [2006]).
It is very important to find the associativity preserving conditions (constructions), where an
associative initial operation µn leads to an associative final operation µ′
n′ during the change of arity.
Example 3-10. An associativity preserving reduction can be given by the construction of a binary
associative operation using (n − 2)-tuple c consisting of nc = n − 2 different constants
µ
(c)
2 [g, h] = µn [g, c, h] . (3.12)
Associativity preserving mixing constructions with different arities and places were considered in
DUDEK AND MICHALSKI [1984], MICHALSKI [1981], SOKHATSKY [1997].
Definition 3-11. An associative polyadic system with identity (3.5) is called a polyadic monoid.
The structure of any polyadic monoid is fixed POP AND POP [2004]: it can be obtained by iterat-
ing a binary operation ˇCUPONA AND TRPENOVSKI [1961] (for polyadic groups this was shown in
D ÖRNTE [1929]).
In polyadic systems, there are several analogs of binary commutativity. The most straightforward
one comes from commutation of the multiplication with permutations.
Definition 3-12. A polyadic system is σ-commutative, if µn = µn ◦ σ, or
µn [g] = µn [σ ◦ g] , (3.13)
where σ◦g = gσ(1), . . . , gσ(n) is a permutated polyad and σ is a fixed element of Sn, the permutation
group on n elements. If (3.13) holds for all σ ∈ Sn, then a polyadic system is commutative.
A special type of the σ-commutativity
µn [g, t, h] = µn [h, t, g] , (3.14)
where t is any fixed (n − 2)-polyad, is called semicommutativity. So for a n-ary semicommutative
system we have
µn g, hn−1
= µn hn−1
, g . (3.15)
If a n-ary semigroup Gsemigrp
is iterated from a commutative binary semigroup with identity, then
Gsemigrp
is semicommutative.
3
This construction is named the b-derived groupoid in DUDEK AND MICHALSKI [1984].

Example 3-13. Let G be the set of natural numbers N, and the 5-ary multiplication is defined by
µ5 [g] = g1 − g2 + g3 − g4 + g5, (3.16)
then GN
5 = N, µ5 is a semicommutative 5-ary monoid having the identity eg = µ5 [g5
] = g for each
g ∈ N. Therefore, GN
5 is the idempotent monoid.
Another possibility is to generalize the binary mediality in semigroups
(g11 · g12) · (g21 · g22) = (g11 · g21) · (g12 · g22) , (3.17)
which, obviously, follows from binary commutativity. But for n-ary systems they are different. It is
seen that the mediality should contain (n + 1) multiplications, it is a relation between n×n elements,
and therefore can be presented in a matrix from. The latter can be achieved by placing the arguments
of the external multiplication in a column.
Definition 3-14. A polyadic system is medial (or entropic), if EVANS [1963], BELOUSOV [1972]
µn


µn [g11, . . . , g1n]
...
µn [gn1, . . . , gnn]

 = µn


µn [g11, . . . , gn1]
...
µn [g1n, . . . , gnn]

 . (3.18)
For polyadic semigroups we use the notation (2.3) and can present the mediality as follows
µn
n [G] = µn
n GT
, (3.19)
where G = gij is the n × n matrix of elements and GT
is its transpose. The semicommuta-
tive polyadic semigroups are medial, as in the binary case, but, in general (except n = 3) not
vice versa GŁAZEK AND GLEICHGEWICHT [1982a]. A more general concept is σ-permutability
STOJAKOVI Ć AND DUDEK [1986], such that the mediality is its particular case with σ = (1, n).
Definition 3-15. A polyadic system is cancellative, if
µn [g, t] = µn [h, t] =⇒ g = h, (3.20)
where g, h can be on any place. This means that the mapping µn is one-to-one in each variable. If
g, h are on the same i-th place on both sides, the polyadic system is called i-cancellative.
The left and right cancellativity are 1-cancellativity and n-cancellativity respectively. A right
and left cancellative n-ary semigroup is cancellative (with respect to the same subset).
Definition 3-16. A polyadic system is called (uniquely) i-solvable, if for all polyads t, u and
element h, one can (uniquely) resolve the equation (with respect to h) for the fundamental operation
µn [u, h, t] = g (3.21)
where h can be on any i-th place.
Definition 3-17. A polyadic system which is uniquely i-solvable for all places i is called a n-ary (or
polyadic) quasigroup.
It follows, that, if (3.21) uniquely i-solvable for all places, than
µℓµ
n [u, h, t] = g (3.22)
can be (uniquely) resolved with respect to h on any place.
Definition 3-18. An associative polyadic quasigroup is called a n-ary (or polyadic) group.

10 STEVEN DUPLIJ
The above definition is the most general one, but it is overdetermined. Much work on polyadic
groups was done RUSAKOV [1998] to minimize the set of axioms (solvability not in all places POST
[1940], CELAKOSKI [1977], decreasing or increasing the number of unknowns in determining equa-
tions GAL’MAK [2003]) or construction in terms of additionally defined objects (various analogs of
the identity and sequences ˇUSAN [2003]), as well as using not total associativity, but instead various
partial ones SOKOLOV [1976], SOKHATSKY [1997], YUREVYCH [2001].
In a polyadic group the only solution of (3.21) is called a querelement of g and denoted by ¯g
D ÖRNTE [1929], such that
µn [h, ¯g] = g, (3.23)
where ¯g can be on any place. So, any idempotent g coincides with its querelement ¯g = g. It follows
from (3.23) and (3.6), that the polyad
ng = gn−2
¯g (3.24)
is neutral for any element of a polyadic group, where ¯g can be on any place. If this i-th place is
important, then we write ng;i. The number of relations in (3.23) can be reduced from n (the number
of possible places) to only 2 (when g is on the first and last places D ÖRNTE [1929], TIMM [1972],
or on some other 2 places). In a polyadic group the Dörnte relations
µn [g, nh;i] = µn [nh;j, g] = g (3.25)
hold true for any allowable i, j. In the case of a binary group the relations (3.25) become g · h · h−1
=
h · h−1
· g = g.
The relation (3.23) can be treated as a definition of the unary queroperation
¯µ1 [g] = ¯g, (3.26)
such that the diagram
G×n µn
✲ G
✑
✑
✑
✑
✑
Prn
✸
G×n
id×(n−1)
×¯µ1
✻
(3.27)
commutes. Then, using the queroperation (3.26) one can give a diagrammatic definition of a polyadic
group (cf. GLEICHGEWICHT AND GŁAZEK [1967]).
Definition 3-19. A polyadic group is a universal algebra
Ggrp
n = G | µn, ¯µ1 | associativity, Dörnte relations , (3.28)
where µn is a n-ary associative operation and ¯µ1 is the queroperation, such that the following diagram
G×(n) id×(n−1)
×¯µ1
✲ G×n ✛ ¯µ1×id×(n−1)
G×n
G × G
id ×Diag(n−1)
✻
Pr1
✲ G
µn
❄
✛ Pr2
G × G
Diag(n−1)×id
✻
(3.29)
commutes, where ¯µ1 can be only on first and second places from the right (resp. left) on the left (resp.
right) part of the diagram.
A straightforward generalization of the queroperation concept and corresponding definitions can
be made by substituting in the above formulas (3.23)–(3.26) the n-ary multiplication µn by iterating
the multiplication µ
ℓµ
n (2.3) (cf. DUDEK [1980] for ℓµ = 2).

Definition 3-20. Let us define the querpower k of g recursively
¯g k
= (¯g k−1 ), (3.30)
where ¯g 0
= g, ¯g 1
= ¯g, or as the k composition ¯µ◦k
1 =
k
¯µ1 ◦ ¯µ1 ◦ . . . ◦ ¯µ1 of the queroperation
(3.26).
For instance GAL’MAK [2003], ¯µ◦2
1 = µn−3
n , such that for any ternary group ¯µ◦2
1 = id, i.e. one has
¯g = g. Using the queroperation in polyadic groups we can define the negative polyadic power
of an element g by the following recursive relation
µn g ℓµ−1
, gn−2
, g −ℓµ
= g, (3.31)
or (after use of (3.2)) as a solution of the equation
µℓµ
n gℓµ(n−1)
, g −ℓµ
= g. (3.32)
It is known that the querpower and the polyadic power are mutually connected DUDEK [1993].
Here, we reformulate this connection using the so called Heine numbers HEINE [1878] or q-deformed
numbers KAC AND CHEUNG [2002]
[[k]]q =
qk
− 1
q − 1
, (3.33)
which have the “nondeformed” limit q → 1 as [k]q → k. Then
¯g k
= g −[[k]]2−n , (3.34)
which can be treated as follows: the querpower coincides with the negative polyadic deformed power
with a “deformation” parameter q which is equal to the “deviation” (2 − n) from the binary group.
4. HOMOMORPHISMS OF POLYADIC SYSTEMS
Let Gn = G; µn and G′
n′ = G′
; µ′
n′ be two polyadic systems of any kind (quasigroup, semi-
group, group, etc.). If they have the multiplications of the same arity n = n′
, then one can define
the mappings from Gn to G′
n. Usually such polyadic systems are similar, and we call mappings
between them the equiary mappings.
Let us take n + 1 mappings ϕGG′
i : G → G′
, i = 1, . . . , n + 1. An ordered system of mappings
ϕGG′
i is called a homotopy from Gn to G′
n, if BELOUSOV [1972]
ϕGG′
n+1 (µn [g1, . . . , gn]) = µ′
n ϕGG′
1 (g1) , . . . , ϕGG′
n (gn) , gi ∈ G. (4.1)
In general, one should add to this definition the “mapping” of the multiplications
µn
ψ
(µµ′
)
nn′
→ µ′
n′ . (4.2)
In such a way, homotopy can be defined as the extended system of mappings ϕGG′
i ; ψ
(µµ′)
nn . The
corresponding commutative (equiary) diagram is
G
ϕGG′
n+1
✲ G′
.............ψ
(µ)
nn
..............✲
G×n
µn
✻
ϕGG′
1 ×...×ϕGG′
n
✲ (G′
)
×n
µ′
n
✻
(4.3)

12 STEVEN DUPLIJ
The existence of the additional “mapping” ψ
(µµ′)
nn acting on the second component of G; µn is
tacitly implied. We will write/mention the “mappings” ψ
(µµ′)
nn′ manifestly, e.g.,
Gn
ϕGG′
i ;ψ
(µµ′
)
nn
⇒ G′
n′ , (4.4)
only as needed. If all the components ϕGG′
i of a homotopy are bijections, it is called an isotopy. In
case of polyadic quasigroups BELOUSOV [1972] all mappings ϕGG′
i are usually taken as permutations
of the same underlying set G = G′
. If the multiplications are also coincide µn = µ′
n, then ϕGG
i ; id is
called an autotopy of the polyadic system Gn. Various properties of homotopy in universal algebras
were studied, e.g. in PETRESCU [1977], HALAˇS [1994].
The homotopy, isotopy and autotopy are widely used equiary mappings in the study of polyadic
quasigroups and loops, while their diagonal equiary counterparts (all ϕGG′
i coincide), the homomor-
phism, isomorphism and automorphism, are more suitable in investigation of polyadic semigroups,
groups and rings and their wide applications in physics. Usually, it is written about the latter between
similar (equiary) polyadic systems: they “...are so well known that we shall not bother to define them
carefully” HOBBY AND MCKENZIE [1988]. Nevertheless, we give a diagrammatic definition of the
standard homomorphism between similar polyadic systems in our notation, which will be convenient
to explain the clear way of its generalization.
A homomorphism from Gn to G′
n is given, if there exists a mapping ϕGG′
: G → G′
satisfying
ϕGG′
(µn [g1, . . . , gn]) = µ′
n ϕGG′
(g1) , . . . , ϕGG′
(gn) , gi ∈ G, (4.5)
which means that the corresponding (equiary) diagram is commutative
G
ϕGG′
✲ G′
.......ψ
(µµ′
)
nn
.........✲
G×n
µn
✻
ϕGG′ ×n
✲ (G′
)
×n
µ′
n
✻
(4.6)
Usually the homomorphism is denoted by the same one letter ϕGG′
, while it would be more consis-
tent to use for its notation the extended pair of mappings ϕGG′
; ψ
(µµ′)
nn . We will use both notations
on a par.
We first mention a small subset of known generalizations of the homomorphism (for bibliography
till 1982 see, e.g., GŁAZEK AND GLEICHGEWICHT [1982b]) and then introduce a concrete con-
struction for an analogous mapping which can change the arity of the multiplication (fundamental
operation) without introducing additional (term) operations. A general approach to mappings be-
tween free algebraic systems was initiated in FUJIWARA [1959], where the so-called basic mapping
formulas for generators were introduced, and its generalization to many-sorted algebras was given in
VIDAL AND TUR [2010]. In NOVOTN Ý [2002] it was shown that the construction of all homomor-
phisms between similar polyadic systems can be reduced to some homomorphisms between corre-
sponding mono-unary algebras NOVOTN Ý [1990]. The notion of n-ary homomorphism is realized as
a sequence of n consequent homomorphisms ϕi, i = 1, . . . , n, of n similar polyadic systems
n
Gn
ϕ1
→ G′
n
ϕ2
→ . . .
ϕn−1
→ G′′
n
ϕn
→ G′′′
n (4.7)
(generalizing Post’s n-adic substitutions POST [1940]) was introduced in GAL’MAK [1998], and
studied in GAL’MAK [2001a, 2007].

The above constructions do not change the arity of polyadic systems, because they are based on the
corresponding diagram which gives a definition of an equiary mapping. To change arity one has to:
1) add another equiary diagram with additional operations using the same formula (4.5), where
both do not change arity;
2) use one modified (and not equiary) diagram and the underlying formula (4.5) by themselves,
which will allow us to change arity without introducing additional operations.
The first way leads to the concept of weak homomorphism which was introduced in
GOETZ [1966], MARCZEWSKI [1966], GŁAZEK AND MICHALSKI [1974] for non-indexed al-
gebras and in GŁAZEK [1980] for indexed algebras, then developed in TRACZYK [1965] for
Boolean and Post algebras, in DENECKE AND WISMATH [2009] for coalgebras and F-algebras
DENECKE AND SAENGSURA [2008] (see also CHUNG AND SMITH [2008]). To define the weak
homomorphism in our notation we should incorporate into the polyadic systems G; µn and G′
; µ′
n′
the following additional term operations of opposite arity νn′ : G×n′
→ G and ν′
n : G′×n
→ G′
and
consider two equiary mappings between G; µn, νn′ and G′
; µ′
n′ , ν′
n .
A weak homomorphism from G; µn, νn′ to G′
, µ′
n′, ν′
n is given, if there exists a mapping ϕGG′
:
G → G′
satisfying two relations simultaneously
ϕGG′
(µn [g1, . . . , gn]) = ν′
n ϕGG′
(g1) , . . . , ϕGG′
(gn) , (4.8)
ϕGG′
(νn′ [g1, . . . , gn′ ]) = µ′
n′ ϕGG′
(g1) , . . . , ϕGG′
(gn′ ) , (4.9)
which means that two equiary diagrams commute
G
ϕGG′
✲ G′
.......ψ
(µν′
)
nn
.........✲
G×n
µn
✻
ϕGG′ ×n
✲ (G′
)
×n
ν′
n
✻
G
ϕGG′
✲ G′
.........ψ
(νµ′)
n′n′
..........✲
G×n′
νn′
✻
ϕGG′ ×n′
✲ (G′
)
×n′
µ′
n′
✻
(4.10)
If only one of the relations (4.8) or (4.9) holds, such a mapping is called a semi-weak
homomorphism KOLIBIAR [1984]. If ϕGG′
is bijective, then it defines a weak isomorphism.
Any weak epimorphism can be decomposed into a homomorphism and a weak isomorphism
GŁAZEK AND MICHALSKI [1977], and therefore the study of weak homomorphisms reduces to
weak isomorphisms (see also CZ ÁK ÁNY [1962], MAL’TCEV [1957], MAL’TSEV [1958]).
5. MULTIPLACE MAPPINGS OF POLYADIC SYSTEMS AND HETEROMORPHISMS
Let us turn to the second way of changing the arity of the multiplication and use only one relation
which we then modify in some natural manner. First, recall that in any set G there always exists
the additional distinguished mapping, viz. the identity idG. We use the multiplication µn with its
combination of idG. We define an (ℓid-intact) id-product for the polyadic system G; µn as
µ(ℓid)
n = µn × (idG)×ℓid
, (5.1)
µ(ℓid)
n : G×(n+ℓid)
→ G×(1+ℓid)
. (5.2)
To indicate the exact i-th place of µn in the r.h.s. of (5.1), we write µ
(ℓid)
n (i), as needed. Here we
use the id-product to generalize the homomorphism and consider mappings between polyadic systems
of different arity. It follows from (5.2) that, if the image of the id-product is G alone, than ℓid = 0.

14 STEVEN DUPLIJ
Let us introduce a multiplace mapping Φ
(n,n′)
k acting as follows
Φ
(n,n′)
k : G×k
→ G′
. (5.3)
While constructing the corresponding diagram, we are allowed to take only one upper Φ
(n,n′)
k ,
because of one G′
in the upper right corner. Since we already know that the lower right corner
is exactly G′×n′
(as a pre-image of one multiplication µ′
n′ ), the lower horizontal arrow should be a
product of n′
multiplace mappings Φ
(n,n′)
k . So we can write a definition of a multiplace analog of
homomorphisms which changes the arity of the multiplication using one relation.
Definition 5-1. A k-place heteromorphism from Gn to G′
n′ is given, if there exists a k-place map-
ping Φ
(n,n′)
k (5.3) such that the following (arity changing or unequiary) diagram is commutative
G×k Φk
✲ G′
G×kn′
µ
(ℓid)
n
✻
(Φk)×n′
✲ (G′
)
×n′
µ′
n′
✻
(5.4)
and the corresponding defining equation (a modification of (4.5)) depends on the place i of µn in
(5.1).
For i = 1 a heteromorphism is defined by the formula
Φ
(n,n′)
k




µn [g1, . . . , gn]
gn+1
...
gn+ℓid



 = µ′
n′

Φ
(n,n′)
k


g1
...
gk

 , . . . , Φ
(n,n′)
k


gk(n′−1)
...
gkn′



 . (5.5)
The notion “heteromorphism” is motivated by ELLERMAN [2006, 2007], where mappings between
objects from different categories were considered and called “chimera morphisms”. See, also, PÉCSI
[2011].
In the particular case n = 3, n′
= 2, k = 2, ℓid = 1 we have
Φ
(3,2)
2
µ3 [g1, g2, g3]
g4
= µ′
2 Φ
(3,2)
2
g1
g2
, Φ
(3,2)
2
g3
g4
. (5.6)
This formula was used in the construction of the bi-element representations of ternary groups
BOROWIEC ET AL. [2006]. Consider the example.
Example 5-2. Let G = Madiag
2 (K), a set of antidiagonal 2 × 2 matrices over the field K and G′
= K,
where K = R, C, Q, H. The ternary multiplication µ3 is a product of 3 matrices. Obviously, µ3 is
not derived from a binary multiplication. For the elements gi =
0 ai
bi 0
, i = 1, 2, we construct a
2-place mapping G × G → G′
as
Φ
(3,2)
2
g1
g2
= a1a2b1b2, (5.7)
which is a heteromorphism, because it satisfies (5.6). Let us introduce a standard 1-place mapping by
ϕ (gi) = aibi, i = 1, 2. It is important to note, that ϕ (gi) is not a homomorphism, because the product
g1g2 belongs to diagonal matrices. Consider the product of mappings
ϕ (g1) · ϕ (g2) = a1b1a2b2, (5.8)

where the product (·) in l.h.s. is taken in K. We observe that (5.7) and (5.8) coincide for the commuta-
tive field K only (= R, C) only, and in this case we can have the relation between the heteromorhism
Φ
(3,2)
2 and the 1-place mapping ϕ
Φ
(3,2)
2
g1
g2
= ϕ (g1) · ϕ (g2) , (5.9)
while for the noncommutative field K (= Q or H) there is no such relation.
A heteromorphism is called derived, if it can be expressed through a 1-place mapping (not nec-
essary a homomorphism). So, in the above Example 5-2 the heteromorphism is derived (by formula
(5.9)) for the commutative field K and nonderived for the noncommutative K.
For arbitrary n a slightly modified construction (5.6) with still binary final arity, defined by n′
= 2,
k = n − 1, ℓid = n − 2,
Φ
(n,2)
n−1




µn [g1, . . . , gn−1, h1]
h2
...
hn−1



 = µ′
2

Φ
(n,2)
n−1


g1
...
gn−1

 , Φ
(n,2)
n−1


h1
...
hn−1



 . (5.10)
was used in DUDEK [2007] to study representations of n-ary groups. However, no new results com-
pared with BOROWIEC ET AL. [2006] (other than changing 3 to n in some formulas) were obtained.
This reflects the fact that a major role is played by the final arity n′
and the number of n-ary multipli-
cations in the l.h.s. of (5.6) and (5.10). In the above cases, the latter number was one, but can make it
arbitrary below n.
Definition 5-3. A heteromorphism is called a ℓµ-ple heteromorphism, if it contains ℓµ multiplica-
tions in the argument of Φ
(n,n′)
k in its defining relation.
According this definition the mapping defined by (5.5) is the 1µ-ple heteromorphism. So by analogy
with (5.1)–(5.2) we define a ℓµ-ple ℓid-intact id-product for the polyadic system G; µn as
µ(ℓµ,ℓid)
n = (µn)×ℓµ
× (idG)×ℓid
, (5.11)
µ(ℓµ,ℓid)
n : G×(nℓµ+ℓid)
→ G×(ℓµ+ℓid)
. (5.12)
Definition 5-4. A ℓµ-ple k-place heteromorphism from Gn to G′
n′ is given, if there exists a k-
place mapping Φ
(n,n′)
k (5.3) such that the following unequiary diagram is commutative
G×k Φk
✲ G′
G×kn′
µ
(ℓµ,ℓid)
n
✻
(Φk)×n′
✲ (G′
)
×n′
µ′
n′
✻
(5.13)
The corresponding main heteromorphism equation is
Φ
(n,n′)
k










µn [g1, . . . , gn] ,
...
µn gn(ℓµ−1), . . . , gnℓµ



ℓµ
gnℓµ+1,
...
gnℓµ+ℓid



ℓid










= µ′
n′

Φ
(n,n′)
k


g1
...
gk

 , . . . , Φ
(n,n′)
k


gk(n′−1)
...
gkn′



 .
(5.14)

16 STEVEN DUPLIJ
FIGURE 1. Dependence of the final arity n′ through the number of heteromorphism places
k for the initial arity n = 9 with the fixed number of intact elements ℓid (left) and the fixed
number of multiplications ℓµ (right): =1 (solid curves), =2 (dash curves)
Obviously, we can consider various permutations of the multiplications on both sides, as further
additional demands (associativity, commutativity, etc.), are introduced, which will be considered be-
low. The commutativity of the diagram (5.13) leads to the system of equation connecting initial and
final arities
kn′
= nℓµ + ℓid, (5.15)
k = ℓµ + ℓid. (5.16)
Excluding ℓµ or ℓid, we obtain two arity changing formulas, respectively
n′
= n −
n − 1
k
ℓid, (5.17)
n′
=
n − 1
k
ℓµ + 1, (5.18)
where n−1
k
ℓid ≥ 1 and n−1
k
ℓµ ≥ 1 are integer.
As an example, the dependences n′
(k) for the fixed ℓµ = 1, 2 and ℓid = 1, 2 with n = 9 are
presented on Figure 1.
The following inequalities hold valid
1 ≤ ℓµ ≤ k, (5.19)
0 ≤ ℓid ≤ k − 1, (5.20)
ℓµ ≤ k ≤ (n − 1) ℓµ, (5.21)
2 ≤ n′
≤ n, (5.22)
which are important for the further classification of heteromorphisms. The main statement follows
from (5.22)
Proposition 5-5. The heteromorphism Φ
(n,n′)
k defined by the general relation (5.14) always decreases
the arity of polyadic multiplication.

Another important observation is the fact that only the id-product (5.11) with ℓid = 0 leads to a
change of the arity. In the extreme case, when k approaches its minimum, k = kmin = ℓµ, the final
arity approaches its maximum n′
max = n, and the id-product becomes a product of ℓµ initial multi-
plications µn without id’s, since now ℓid = 0 in (5.14). Therefore, we call a heteromorphism defined
by (5.14) with ℓid = 0 a k (= ℓµ)-place homomorphism. The ordinary homomorphism (4.1) corre-
sponds to k = ℓµ = 1, and so it is really a 1-place homomorphism. An opposite extreme case, when
the final arity approaches its minimum n′
min = 2 (the final operation is binary), corresponds to the
maximal value of k, that is k = kmax = (n − 1) ℓµ. The number of id’s now is ℓid = (n − 2) ℓµ ≥ 0,
which vanishes, when the initial operation is binary as well. This is the case of the ordinary homo-
morphism (4.1) for both binary operations n′
= n = 2 and k = ℓµ = 1. We conclude that:
Any polyadic system can be mapped into a binary system by means of the special k-place ℓµ-ple
heteromorphism Φ
(n,n′)
k , where k = (n − 1) ℓµ (we call it a binarizing heteromorphism) which
is defined by (5.14) with ℓid = (n − 2) ℓµ.
In relation to the Gluskin-Hosszú theorem GLUSKIN [1965] (any n-ary group can be constructed
from the special binary group and its homomorphism) our statement can be treated as:
Theorem 5-6. Any n-ary system can be mapped into a binary system, using a suitable binarizing
heteromorphism Φ
(n,2)
k (5.14).
The case of 1-ple binarizing heteromorphism (ℓµ = 1) corresponds to the formula (5.10). Further
requirements (associativity, commutativity, etc.) will give additional relations between multiplications
and Φ
(n,n′)
k , and fix the exact structure of (5.14). Thus, we arrive to the following
Proposition 5-7. Classification of ℓµ-ple heteromorphisms:
(1) n′
= n′
max = n =⇒ Φ
(n,n)
k is the ℓµ-place or multiplace homomorphism, i.e.,
k = kmin = ℓµ. (5.23)
(2) 2 < n′
< n =⇒ Φ
(n,n′)
k is the intermediate heteromorphism with
k =
n − 1
n′ − 1
ℓµ. (5.24)
In this case the number of intact elements is proportional to the number of multiplications
ℓid =
n − n′
n′ − 1
ℓµ. (5.25)
(3) n′
= n′
min = 2 =⇒ Φ
(n,2)
k is the (n − 1) ℓµ-place (multiplace) binarizing
heteromorphism, i.e.,
k = kmax = (n − 1) ℓµ. (5.26)
In the extreme (first and third) cases there are no restrictions on the initial arity n, while in the
intermediate case n is “quantized” due to the fact that fractions in (5.17) and (5.18) should be integers.
Observe, that in the extreme (first and third) cases there are no restrictions on the initial arity n,
while in the intermediate case n is “quantized” due to the fact that fractions in (5.17) and (5.18) should
be integer. In this way, we obtain the TABLE 1 for the series of n and n′
(we list only first ones, just
for 2 ≤ k ≤ 4 and include the binarizing case n′
= 2 for completeness).
Thus, we have established a general structure and classification of heteromorphisms defined by
(5.14). The next important issue is the preservation of special properties (associativity, commutativity,
etc.), while passing from µn to µ′
n′ , which will further restrict the concrete shape of the main relation

18 STEVEN DUPLIJ
TABLE 1. “Quantization” of heteromorphisms
k ℓµ ℓid n/n′
2 1 1
n = 3, 5, 7, . . .
n′
= 2, 3, 4, . . .
3 1 2
n = 4, 7, 10, . . .
n′
= 2, 3, 4, . . .
3 2 1
n = 4, 7, 10, . . .
n′
= 3, 5, 7, . . .
4 1 3
n = 5, 9, 13, . . .
n′
= 2, 3, 4, . . .
4 2 2
n = 3, 5, 7, . . .
n′
= 2, 3, 4, . . .
4 3 1
n = 5, 9, 13, . . .
n′
= 4, 7, 10, . . .
(5.14) for each choice of the heteromorphism parameters: arities n, n′
, places k, number of intacts
ℓid and multiplications ℓµ.
6. ASSOCIATIVITY QUIVERS AND HETEROMORPHISMS
The most important property of the heteromorphism, which is needed for its next applications to
representation theory, is the associativity of the final operation µ′
n′ , when the initial operation µn is
associative. In other words, we consider here the concrete form of semigroup heteromorphisms. In
general, this is a complicated task, because it is not clear from (5.14), which permutation in the l.h.s.
should be taken to get an associative product in its r.h.s. for each set of the heteromorphism param-
eters. Straightforward checking of the associativity of the final operation µ′
n′ for each permutation
in the l.h.s. of (5.14) is almost impossible, especially for higher n. To solve this difficulty we intro-
duce the concept of the associative polyadic quiver and special rules to construct the associative final
operation µ′
n′ .
Definition 6-1. A polyadic quiver of products is the set of elements from Gn (presented as
several copies of some matrix of the elements glued together) and arrows, such that the elements
along arrows form n-ary products µn.
For instance, for the 4-ary multiplication µ4 [g1, h2, g2, u1] (elements from Gn are arbitrary here)
a corresponding 4-adic quiver will be denoted by {g1 → h2 → g2 → u1}, and graphically this 4-adic
quiver is
g1
❅❅❅
h1 u1 g1 h1 u1
g2 h2
**
u2 g2
77♥♥♥♥♥♥♥♥♥
h2 u2
corr
KS

µ4 [g1, h2, g2, u1] .
(6.1)
Next we define polyadic quivers which are related to the main heteromorphism equation (5.14)
in the following way:

1) the matrix of elements is the transposed matrix from the r.h.s. of (5.14), such that different letters
correspond to their place in Φ
(n,n′)
k and the low index of an element is related to its position in the µ′
n′
product;
2) the number of polyadic quivers is ℓµ, which corresponds to ℓµ multiplications in the l.h.s. of
(5.14);
3) the heteromorphism parameters (n, n′
, k, ℓid and ℓµ) are not arbitrary, but satisfy the arity
changing formulas (5.17)-(5.18);
4) the intact elements will be placed after a semicolon.
In this way, a polyadic quiver makes a clear visualization of the main heteromorphism equation
(5.14), and later on it will allow us to distinguish associativity preserving heteromorphisms by precise
graphical rules.
For example, the polyadic quiver {g1 → h2 → g2 → u1; h1, u2} corresponds to the unequiary het-
eromorphism with n = 4, n′
= 2, k = 3, ℓid = 2 and ℓµ = 1 is
g1
❅❅❅
h1 u1 g1 h1 u1
g2 h2
**
u2 g2
77♥♥♥♥♥♥♥♥♥
h2 u2
corr
KS

Φ
(4,2)
3


µ4 [g1, h2, g2, u1]
h1
u2

 = µ′
2

Φ
(4,2)
3


g1
h1
u1

 , Φ
(4,2)
3


g2
h2
u2



 ,
(6.2)
where the intact elements h1, u2 are boxed in squares. As it is seen from (6.2), the product µ′
2 is not
associative, if µ4 is associative. So, not all polyadic quivers preserve associativity.
Definition 6-2. An associative polyadic quiver is a polyadic quiver which ensures the final
associativity of µ′
n′ in the main heteromorphism equation (5.14), when the initial multiplication µn is
associative.
One of the associative polyadic quivers which corresponds to the same heteromorphism parameters,
as the non-associative quiver (6.2), is {g1 → h2 → u1 → g2; h1, u2} which corresponds to
g1
❅❅❅
h1 u1
❅❅❅❅ g1 h1 u1
g2 h2
⑥⑥⑥⑥
u2 g2 h2 u2
corr
KS

Φ
(4,2)
3


µ4 [g1, h2, u1, g2]
h1
u2

 = µ′
2

Φ
(4,2)
3


g1
h1
u1

 , Φ
(4,2)
3


g2
h2
u2



 .
(6.3)
Here we propose a classification of associative polyadic quivers and the rules of construction of
the corresponding heteromorphism equations, and then use the heteromorphism parameters for the
classification of ℓµ-ple heteromorphisms (5.24). In other words, we describe a consistent procedure
for building the semigroup heteromorphisms.
Let us consider the first class of heteromorphisms (without intact elements ℓid = 0 or intactless),
that is ℓµ-place (multiplace) homomorphisms. In the simplest case, associativity can be achieved,
when all elements in a product are taken from the same row. The number of places k is not fixed by

20 STEVEN DUPLIJ
the arity relation (5.17) and can be arbitrary, while the arrows can have various directions. There are
2k
such combinations which preserve associativity. If the arrows have the same direction, this kind of
mapping is also called a homomorphism. As an example, for n = n′
= 3, k = 2, ℓµ = 2 we have
g1

h1

g2

h2

g3 h3
corr
KS

Φ
(3,3)
2
µ3 [g1, g2, g3]
µ3 [h1, h2, h3]
= µ′
3 Φ
(3,3)
2
g1
h1
, Φ
(3,3)
2
g2
h2
, Φ
(3,3)
2
g3
h3
.
(6.4)
Note that the analogous quiver with opposite arrow directions is
g1

h1
g2

h2
OO
g3 h3
OO
corr
KS

Φ
(3,3)
2
µ3 [g1, g2, g3]
µ3 [h3, h2, h1]
= µ′
3 Φ
(3,3)
2
g1
h1
, Φ
(3,3)
2
g2
h2
, Φ
(3,3)
2
g3
h3
,
(6.5)
The latter mapping and the corresponding vertical quiver were used in constructing the middle repre-
sentations of ternary groups BOROWIEC ET AL. [2006].
For nonvertical quivers the main rule is the following: all arrows of an associative quiver should
have direction from left to right or vertical, and they should not instersect. Also, we start always from
the upper left corner, because of the permutation symmetry of (5.14), we can rearrange and rename
variables in the necessary way.
An important class of intactless heteromorphisms (with ℓid = 0) preserving associativity can be
constructed using an analogy with the Post substitutions POST [1940], and therefore we call it the
Post-like associative quiver. The number of places k is now ﬁxed by k = n−1, while n′
= n
and ℓµ = k = n − 1. An example of the Post-like associative quiver with the same heteromorphisms
parameters as in (6.4)-(6.5) is
g1
❅❅❅
h1
❅❅❅❅ g1 h1
g2 h2
❅❅❅❅ g2
❅❅❅
h2
g3 h3 g3 h3
corr
KS

Φ
(3,3)
2
µ3 [g1, h2, g3]
µ3 [h1, g2, h3]
= µ′
3 Φ
(3,3)
2
g1
h1
, Φ
(3,3)
2
g2
h2
, Φ
(3,3)
2
g3
h3
.
(6.6)

This construction appeared in the study of ternary semigroups of morphisms CHRONOWSKI
[1994b,a], CHRONOWSKI AND NOVOTN Ý [1995]. Its n-ary generalization was used in the con-
sideration of polyadic operations on Cartesian powers GAL’MAK [2008], polyadic analogs of the
Cayley and Birkhoff theorems GAL’MAK [2001b, 2007] and special representations of n-groups
GLEICHGEWICHT ET AL. [1983], WANKE-JAKUBOWSKA AND WANKE-JERIE [1984] (where the
n-group with the multiplication µ′
2 was called the diagonal n-group). Consider the following ex-
ample.
Example 6-3. Let Λ be the Grassmann algebra consisting of even and odd parts Λ = Λ¯0 ⊕ Λ¯1 (see
e.g., BEREZIN [1987]). The odd part can be considered as a ternary semigroup G
(¯1)
3 = Λ¯1, µ3 ,
its multiplication µ3 : Λ¯1 × Λ¯1 × Λ¯1 → Λ¯1 is defined by µ3 [α, β, γ] = α · β · γ, where (·) is
multiplication in Λ and α, β, γ ∈ Λ¯1, so G
(¯1)
3 is nonderived and contains no unity. The even part can
be treated as a ternary group G
(¯0)
3 = Λ¯0, µ′
3 with the multiplication µ′
3 : Λ¯0 ×Λ¯0 ×Λ¯0 → Λ¯0, defined
by µ3 [a, b, c] = a · b · c, where a, b, c ∈ Λ¯0, thus G
(¯0)
3 is derived and contains unity. We introduce the
heteromorphism G
(¯1)
3 → G
(¯0)
3 as a mapping (2-place homomorphism) Φ
(3,3)
2 : Λ¯1 × Λ¯1 → Λ¯0 by the
formula
Φ
(3,3)
2
α
β
= α · β, (6.7)
where α, β ∈ Λ¯1. It is seen that Φ
(3,3)
2 defined by (6.7) satisfies the Post-like heteromorphism equation
(6.6), but not the “vertical” one (6.4), due to the anticommutativity of odd elements from Λ¯1. In other
words, G
(¯0)
3 can be treated as a nontrivial example of the “diagonal” semigroup of G
(¯1)
3 (according
to the notation of GLEICHGEWICHT ET AL. [1983], WANKE-JAKUBOWSKA AND WANKE-JERIE
[1984]).
Note that for the number of places k ≥ 3 there exist additional (to the above) associative quivers
having the same heteromorphism parameters. For instance, when n′
= n = 4 and k = 3 we have the
Post-like associative quiver
g1
❅❅❅
h1
❆❆❆❆ u1
❅❅❅❅ g1 h1 u1
g2 h2
❆❆❆❆ u2
❅❅❅❅ g2
❅❅❅
h2 u2
g3 h3 u3
❅❅❅❅ g3
❅❅❅
h3
❆❆❆❆ u3
g4 h4 u4 g4 h4 u4
corr
KS

Φ
(4,4)
3


µ4 [g1, h2, u3, g4]
µ4 [h1, u2, g3, h4]
µ4 [u1, g2, h3, u4]

 = µ′
4

Φ
(4,4)
3


g1
h1
u1

 , Φ
(4,4)
3


g2
h2
u2

 , Φ
(4,4)
3


g1
h1
u1

 , Φ
(4,4)
3


g2
h2
u2



 .
(6.8)

22 STEVEN DUPLIJ
Also, we have one intermediate non-Post associative quiver
g1
''PPPPPPPPP h1
''PPPPPPPPP u1
''PPPPPPPP g1 h1 u1 g1 h1 u1
g2 h2 u2
''PPPPPPPP g2
''PPPPPPPPP h2
''PPPPPPPPP u2 g2 h2 u2
g3 h3 u3 g3 h3
''PPPPPPPPP u3
''PPPPPPPP g3
''PPPPPPPPP h3 u3
g4 h4 u4 g4 h4 u4 g4 h4 u4
corr
KS

Φ
(4,4)
3


µ4 [g1, u2, h3, g4]
µ4 [h1, g2, u3, h4]
µ4 [u1, h2, g3, u4]

 = µ′
4

Φ
(4,4)
3


g1
h1
u1

 , Φ
(4,4)
3


g2
h2
u2

 , Φ
(4,4)
3


g1
h1
u1

 , Φ
(4,4)
3


g2
h2
u2



 .
(6.9)
A general method of constructing associative quivers for n′
= n, ℓid = 0 and k = n − 1 can be
illustrated from the following more complicated example with n = 5. First, we draw k = n − 1
(= 4) copies of element matrices. Then we go from the first element in the first column g1 to the last
element in this column gn′ (= g5) by different k = n − 1 (= 4) ways: by the vertical quiver, by the
Post-like quiver (going to the second copy of the element matrix) and by the remaining k − 2 (= 2)
non-Post associative quivers, as
g1
$$❍❍❍❍❍
))❘❘❘❘❘❘❘❘❘❘
**❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ h1 u1 v1 g1 h1 u1 v1 g1 h1 u1 v1 g1 h1 u1 v1
g2

h2
❆❆❆❆ u2
((◗◗◗◗◗◗◗◗◗ v2
**❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ g2 h2 u2 v2 g2 h2 u2 v2 g2 h2 u2 v2
g3

h3 u3
❅❅❅❅ v3 g3
((◗◗◗◗◗◗◗◗◗ h3 u3
++❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ v3 g3 h3 u3 v3 g3 h3 u3 v3
g4

h4 u4 v4
!!❉❉❉❉❉ g4 h4 u4
))❘❘❘❘❘❘❘❘❘❘❘ v4 g4 h4
++❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ u4 v4 g4 h4 u4 v4
g5 h5 u5 v5 g5 h5 u5 v5 g5 h5 u5 v5 g5 h5 u5 v5
vertical Post non-Post non-Post
.
(6.10)
Here we show, for short, only the quiver itself without the corresponding heteromorphism equation
and only the arrows corresponding to the first product, while other arrows (starting from h1, u1, v1)
are parallel to it, as in (6.9).
The next type of heteromorphisms (intermediate) is described by the equations (5.15)-(5.25), it
contains intact elements (ℓid ≥ 1) and changes (decreases) arity n′
n. For each fixed k the arities
are not arbitrary and presented in TABLE 1. The first general rule is: the associative quivers are non-
decreasing in both,vertical (from up to down) and horizontal (from left to right), directions. Second,
if there are several multiplications (ℓµ ≥ 2), the corresponding associative quivers do not intersect.
Let us present some examples and start from the smallest number of heteromorphism places in
Φ
(n,n′)
k . For k = 2, the first (nonbinarizing n′
≥ 3) case is n = 5, n′
= 3, ℓid = 1 (see the first row

and second n/n′
pair of TABLE 1). The corresponding associative quivers are
g1
// h1
❅❅❅❅ g1 h1 g1 h1
g2 h2 g2
// h2
❅❅❅❅ g2 h2
g3 h3 g3 h3 g3 h3
corr
KS

Φ
(5,3)
2
µ5 [g1, h1,g2, h2,g3]
h3
g1
❅❅❅
h1 g1 h1 g1 h1
g2 h2
// g2
❅❅❅
h2 g2 h2
g3 h3 g3 h3
// g3 h3
corr
KS

Φ
(5,3)
2
µ5 [g1, h2,g2, h3,g3]
h1
,
(6.11)
where we do not write the r.h.s. of the heteromorphism equation (5.14), because it is simply related
to the transposed quiver matrix of the size n′
× k.
More complicated examples can be given for k = 3, which corresponds to the second and third lines
of TABLE 1. That is we can obtain a ternary ﬁnal product (n′
= 3) by using one or two multiplications
ℓµ = 1, 2. Examples of the corresponding associative quivers are (ℓµ = 1)
g1
// h1
❆❆❆❆ u1 g1 h1 u1 g1 h1 u1
g2 h2 u2
// g2
❅❅❅
h2 u2 g2 h2 u2
g3 h3 u3 g3 h3
// u3
// g3 h3 u3
corr
KS

Φ
(7,3)
3


µ7 [g1, h1,u2, g2, h3, u3, g3]
h2
u1


(6.12)
and (ℓµ = 2)
g1
''PPPPPPPPP h1 22u1 g1 h1 u1
❅❅❅❅ g1 h1 u1
g2 h2 u2 44g2 h2
''PPPPPPPPP u2 g2
❅❅❅
h2 u2
g3 h3 u3 g3 h3 u3 g3 h3 u3
corr
KS

Φ
(4,3)
3


µ4 [g1, u2, h2, g3]
µ4 [h1,u1, g2, h3]
u3


(6.13)
respectively. Finally, for the case k = 4 one can construct the associative quiver corresponding to
the ﬁrst pair of the last line in TABLE 1. It has three multiplications and one intact element, and the

24 STEVEN DUPLIJ
corresponding quiver is, e.g.,
g1
❅❅❅
h1
--
u1 v1 33g1 h1 u1
''PPPPPPPP v1 g1 h1 u1 v1
g2 h2
❆❆❆❆ u2 v2 33g2 h2 u2
❅❅❅❅ v2 g2
❅❅❅
h2 u2 v2
g3 h3 u3 44v3 g3
++❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ h3 u3 v3
''PPPPPPPP g3 h3
''PPPPPPPPP u3 v3
g4 h4 u4 v4 g4 h4 u4 v4 g4 h4 u4 v4
corr
KS

Φ
(5,4)
4




µ5 [g1, h2, u3, g3, g4]
µ5 [h1, v2, u2, v3, h4]
µ5 [v1, u1, g2, h3, v4]
u4



 .
(6.14)
There are many other possibilities (using permutations and different variants of quivers) to obtain
an associative final product µ′
n′ corresponding the same heteromorphism parameters, and therefore
we do list them all. The above examples are sufficient to understand the rules of the associative quiver
construction and obtain the polyadic semigroup heteromorphisms.
7. MULTIPLACE REPRESENTATIONS OF POLYADIC SYSTEMS
Representation theory (see e.g. KIRILLOV [1976]) deals with mappings from abstract alge-
braic systems into linear systems, such as, e.g. linear operators in vector spaces, or into general
(semi)groups of transformations of some set. In our notation, this means that in the mapping of
polyadic systems (4.4) the final multiplication µ′
n′ is a linear map. This leads to some restrictions on
the final polyadic structure G′
n′ , which are considered below.
Let V be a vector space over a field K (usually algebraically closed) and End V be a set of linear
endomorphisms of V , which is in fact a binary group. In the standard way, a linear representation
of a binary semigroup G2 = G; µ2 is a (1-place) map Π1 : G2 → End V , such that Π1 is a
homomorphism
Π1 (µ2 [g, h]) = Π1 (g) ∗ Π1 (h) , (7.1)
where g, h ∈ G and (∗) is the binary multiplication in End V (usually, it is a (semi)group with
multiplication as composition of operators or product of matrices, if a basis is chosen). If G2 is a
binary group with the unity e, then we have the additional condition
Π1 (e) = idV . (7.2)
We will generalize these known formulas to the corresponding polyadic systems along with the
heteromorphism concept introduced above. Our general idea is to use the heteromorphism equation
(5.14) instead of the standard homomorphism equation (7.1), such that the arity of the representation
will be different from the arity of the initial polyadic system n′
= n.
Consider the structure of the final n′
-ary multiplication µ′
n′ in (5.14), taking into account that the
final polyadic system G′
n′ should be constructed from End V . The most natural and physically appli-
cable way is to consider the binary End V and to put G′
n′ = dern′ (End V ), as it was proposed for the
ternary case in BOROWIEC ET AL. [2006]. In this way G′
n′ becomes a derived n′
-ary (semi)group of
endomorphisms of V with the multiplication µ′
n′ : (End V )×n′
→ End V , where
µ′
n′ [v1, . . . , vn′ ] = v1 ∗ . . . ∗ vn′ , vi ∈ End V. (7.3)

Because the multiplication µ′
n′ (7.3) is derived and is therefore associative by definition, we may
consider the associative initial polyadic systems (semigroups and groups) and the associativity pre-
serving mappings that are the special heteromorphisms constructed in the previous section.
Let Gn = G; µn be an associative n-ary polyadic system. By analogy with (5.3), we introduce
the following k-place mapping
Π
(n,n′)
k : G×k
→ End V. (7.4)
A multiplace representation of an associative polyadic system Gn in a vector space V is
given, if there exists a k-place mapping (7.4) which satisfies the (associativity preserving) heteromor-
phism equation (5.14), that is
Π
(n,n′)
k










µn [g1, . . . , gn] ,
...



ℓµ
gnℓµ+1,
...
gnℓµ+ℓid



ℓid










=
n′
Π
(n,n′)
k


g1
...
gk

 ∗ . . . ∗ Π
(n,n′)
k


gk(n′−1)
...
gkn′

, (7.5)
and the following diagram commutes
G×k Πk
✲ End V
G×kn′
µ
(ℓµ,ℓid)
n
✻
(Πk)×n′
✲ (End V )×n′
(∗)n′
✻
(7.6)
where µ
(ℓµ,ℓid)
n is given by (5.11), ℓµ and ℓid are the numbers of multiplications and intact elements in
the l.h.s. of (7.5), respectively.
The exact permutation in the l.h.s. of (7.5) is given by the associative quiver presented in the
previous section. The representation parameters (n, n′
, k, ℓµ and ℓid) in (7.5) are the same as the
heteromorphism parameters, and they satisfy the same arity changing formulas (5.17) and (5.18).
Therefore, a general classification of multiplace representations can be done by analogy with that of
the heteromorphisms (5.23)–(5.26) as follows:
(1) The hom-like multiplace representation which is a multiplace homomorphism with
n′
= n′
max = n, without intact elements lid = l
(min)
id = 0, and minimal number of places
k = kmin = ℓµ. (7.7)
(2) The intact element multiplace representation which is the intermediate heteromor-
phism with 2 n′
n and the number of intact elements is
lid =
n − n′
n′ − 1
ℓµ. (7.8)
(3) The binary multiplace representation which is a binarizing heteromorphism (5.26)
with n′
= n′
min = 2, the maximal number of intact elements l
(max)
id = (n − 2) ℓµ and maximal
number of places
k = kmax = (n − 1) ℓµ. (7.9)

26 STEVEN DUPLIJ
The multiplace representations for n-ary semigroups have no additional defining relations, as com-
pared with (7.5). In case of n-ary groups, we need an analog of the “normalizing” relation (7.2). If
the n-ary group has the unity e, then one can put
Π
(n,n′)
k


e
...
e



k

 = idV . (7.10)
If there is no unity at all, one can “normalize” the multiplace representation, using analogy with (7.2)
in the form
Π1 h−1
∗ h = idV , (7.11)
as follows
Π
(n,n′)
k









¯h
...
¯h



ℓµ
h
...
h



ℓid









= idV , (7.12)
for all h ∈Gn, where ¯h is the querelement of h. The latter ones can be placed on any places in the
l.h.s. of (7.12) due to the Dörnte identities. Also, the multiplications in the l.h.s. of (7.5) can change
their place due to the same reason.
A general form of multiplace representations can be found by applying the Dörnte identities to each
n-ary product in the l.h.s. of (7.5). Then, using (7.12) we have schematically
Π
(n,n′)
k


g1
...
gk

 = Π
(n,n′)
k









t1
...
tℓµ
g
...
g



ℓid









, (7.13)
where g is an arbitrary fixed element of the n-ary group and
ta = µn [ga1, . . . , gan−1, ¯g] , a = 1, . . . , ℓµ. (7.14)
This is the special shape of some multiplace representations, while the concrete formulas should be
obtained in each case separately. Nevertheless, some conclusions can be drawn from (7.13). Firstly,
the equivalence classes on which Π
(n,n′)
k is constant are determined by fixing ℓµ + 1 elements, i.e. by
the surface ta = const, g = const. Secondly, some k-place representations of a n-ary group can be
reduced to ℓµ-place representations of its retract. In the case ℓµ = 1, multiplace representations of
a n-ary group derived from a binary group correspond to ordinary representations of the latter (see
BOROWIEC ET AL. [2006], DUDEK [2007]).

Example 7-1. Let us consider the case of a binary multiplace representation Π
(n,2)
2n−2 of n-ary group
Gn = G; µn with two multiplications ℓµ = 2 defined by the associativity preserving equation
Π
(n,2)
2n−2















µn [g1, u1, . . . un−2, g2]
u′
1
...
u′
n−2
µn [h1, v1, . . . vn−2, h2]
v′
1
...
v′
n−2















= Π
(n,2)
2n−2















g1
u1
...
un−2
h1
v1
...
vn−2















∗ Π
(n,2)
2n−2















g2
u′
1
...
u′
n−2
h2
v′
1
...
v′
n−2















(7.15)
and normalizing condition
Π
(n,2)
2n−2















h
...
h



n − 2
¯h
h
...
h



n − 2
¯h















= idV , (7.16)
where h ∈Gn is arbitrary. Using (7.15) and (7.16) for a general form of this (2n − 2)-place repre-
sentation we have
Π
(n,2)
2n−2 = Π
(n,2)
2n−2















g1
u1
...
un−2
h1
v1
...
vn−2















= Π
(n,2)
2n−2















g1
u1
...
un−2
h1
v1
...
vn−2















∗ Π
(n,2)
2n−2















h
...
h



n − 2
¯h
h
...
h



n − 2
¯h















= Π
(n,2)
2n−2



















µn [g1, u1, . . . un−2, h]
h
...
h



n − 3
¯h
µn [h1, v1, . . . vn−2, h]
h
...
h



n − 3
¯h



















.
(7.17)
The Dörnte identity applied to the first elements of the products in the r.h.s. of (7.17) together with
associativity of µn gives
Π
(n,2)
2n−2 = Π
(n,2)
2n−2



























µn


n−2
g, . . . , g, tg, h


h
...
h



n − 3
¯h
µn


n−2
g, . . . , g, th, h


h
...
h



n − 3
¯h



























= Π
(n,2)
2n−2















g
...
g



n − 2
tg
g
...
g



n − 2
th















∗Π
(n,2)
2n−2















h
...
h



n − 2
¯h
h
...
h



n − 2
¯h















= Π
(n,2)
2n−2















g
...
g



n − 2
tg
g
...
g



n − 2
th















,
(7.18)

28 STEVEN DUPLIJ
where
tg = µn [¯g, g1, u1, . . . un−2] , th = µn [¯g, h1, v1, . . . vn−2] . (7.19)
Thus, the equivalent classes of the multiplace representation Π
(n,2)
2n−2 (7.15) are determined by the
(ℓµ + 1 = 3)-element surface
tg = const, th = const, g = const. (7.20)
Note that the “ℓµ-place reduction” of a multiplace representation is possible not for all associativity
preserving heteromorphism equations. For instance, if to exchange vi ↔ v′
i in l.h.s. of (7.15), then
the associativity remains, but the “ℓµ-place reduction”, analogous to (7.18) will not be possible.
In case, when it is possible, the corresponding ℓµ-place representation can be realized on the binary
retract of Gn (for some special n-ary groups and ℓµ = 1 see see BOROWIEC ET AL. [2006], DUDEK
[2007]). Indeed, let Gret
2 = G, ⊛ = retg G; µn , where g1 ⊛g2
def
= µn g1,
n−2
g, . . . , g, g2 , g1, g2, g ∈
G and we deﬁne the reduced ℓµ-place representation (now ℓµ = 2) through (7.18) as follows
Πret g
2
g1
g2
def
= Π
(n,2)
2n−2















g
...
g



n − 2
g1
g
...
g



n − 2
g2















. (7.21)
From (7.17) and (7.16) we obtain
Πret g
2
g1
g2
∗ Πret g
2
g′
1
g′
2
= Π
(n,2)
2n−2















g
...
g



n − 2
g1
g
...
g



n − 2
g2















∗ Π
(n,2)
2n−2















g
...
g



n − 2
g′
1
g
...
g



n − 2
g′
2















= Π
(n,2)
2n−2



























µn


n−2
g, . . . , g, g1, g


g
...
g



n − 3
g′
1
µn


n−2
g, . . . , g, g2, g


g
...
g



n − 3
g′
2



























= Π
(n,2)
2n−2






















g
...
g



n − 2
µn

g1,
n−2
g, . . . , g, g′
1


g
...
g



n − 2
µn

g2,
n−2
g, . . . , g, g′
2
























= Π
(n,2)
2n−2















g
...
g



n − 2
g1 ⊛ g′
1
g
...
g



n − 2
g2 ⊛ g′
2















= Πret g
2
g1 ⊛ g′
1
g2 ⊛ g′
2
.

In the framework of our classification Πret g
2
g1
g2
is a hom-like 2-place (binary) representation.
The above formulas describe various properties of multiplace representations, but they give no
idea of how to build representations for concrete polyadic systems. The most common method of
representation construction uses the concept of a group action on a set (see, e.g., KIRILLOV [1976]).
Below we extend this concept to the multiplace case and use corresponding heteromorphisms, as it
was done above for homomorphisms and representations.
8. MULTIACTIONS AND G-SPACES
Let Gn = G; µn be a polyadic system and X be a set. A (left) 1-place action of Gn on X is the
external binary operation ρ
(n)
1 : G × X → X such that it is consistent with the multiplication µn, i.e.
composition of the binary operations ρ1 {g|x} gives the n-ary product, that is,
ρ
(n)
1 {µn [g1, . . . gn] |x} = ρ
(n)
1 g1|ρ
(n)
1 g2| . . . |ρ
(n)
1 {gn|x} . . . , g1, . . . , gn ∈ G, x ∈ X. (8.1)
If the polyadic system is a n-ary group, then in addition to (8.1) it is implied the there exist such
ex ∈ G (which may or may not coincide with the unity of Gn) that ρ
(n)
1 {ex|x} = x for all x ∈ X,
and the mapping x → ρ
(n)
1 {ex|x} is a bijection of X. The right 1-place actions of Gn on X are
defined in a symmetric way, and therefore we will consider below only one of them. Obviously, we
cannot compose ρ
(n)
1 and ρ
(n′)
1 with n = n′
. Usually X is called a G-set or G-space depending on its
properties (see, e.g., HUSEM ÖLLER ET AL. [2008]).
The application of the 1-place action defined by (8.1) to the representation theory of n-ary groups
gave mostly repetitions of the ordinary (binary) group representation results (except for trivial b-
derived ternary groups) DUDEK AND SHAHRYARI [2012]. Also, it is obviously seen that the con-
struction (8.1) with the binary external operation ρ1 cannot be applied for studying the most important
regular representations of polyadic systems, when the X coincides with Gn itself and the action arises
from translations.
Here we introduce the multiplace concept of action for polyadic systems, which is consistent with
heteromorphisms and multiplace representations. Then we will show how it naturally appears when
X = Gn and apply it to construct examples of representations including the regular ones.
For a polyadic system Gn = G; µn and a set X we introduce an external polyadic operation
ρk : G×k
× X → X, (8.2)
which is called a (left) k-place action or multiaction. To generalize the 1-action composition
(8.1), we use the analogy with multiplication laws of the heteromorphisms (5.14) and the multiplace
representations (7.5) and propose (schematically)
ρ
(n)
k



µn [g1, . . . , gn] ,
...



ℓµ
gnℓµ+1,
...
gnℓµ+ℓid



ℓid
x



= ρ
(n)
k
n′



g1
...
gk
. . . ρ
(n)
k



gk(n′−1)
...
gkn′
x



. . .



. (8.3)
The connection between all the parameters here is the same as in the arity changing formulas
(5.17)–(5.18). Composition of mappings is associative, and therefore in concrete cases we can use

30 STEVEN DUPLIJ
the associative quiver technique, as it is described in the previous sections. If Gn is n-ary group, then
we should add to (8.3) the “normalizing” relations analogous with (7.10) or (7.12). So, if there is a
unity e ∈Gn, then
ρ
(n)
k



e
...
e
x



= x, for all x ∈ X. (8.4)
In terms of the querelement, the normalization has the form
ρ
(n)
k



¯h
...
¯h



ℓµ
h
...
h



ℓid
x



= x, for all x ∈ X and for all h ∈ Gn. (8.5)
The multiaction ρ
(n)
k is transitive, if any two points x and y in X can be “connected” by ρ
(n)
k , i.e.
there exist g1, . . . , gk ∈Gn such that
ρ
(n)
k



g1
...
gk
x



= y. (8.6)
If g1, . . . , gk are unique, then ρ
(n)
k is sharply transitive. The subset of X, in which any points
are connected by (8.6) with fixed g1, . . . , gk can be called the multiorbit of X. If there is only one
multiorbit, then we call X the heterogenous G-space (by analogy with the homogeneous one). By
analogy with the (ordinary) 1-place actions, we define a G-equivariant map Ψ between two G-sets X
and Y by (in our notation)
Ψ

ρ
(n)
k



g1
...
gk
x




 = ρ
(n)
k



g1
...
gk
Ψ (x)



∈ Y, (8.7)
which makes G-space into a category (for details, see, e.g., HUSEM ÖLLER ET AL. [2008]). In the
particular case, when X is a vector space over K, the multiaction (8.2) can be called a multi-G-
module which satisfies (8.4) and the additional (linearity) conditions
ρ
(n)
k



g1
...
gk
ax + by



= aρ
(n)
k



g1
...
gk
x



+ bρ
(n)
k



g1
...
gk
y



, (8.8)
where a, b ∈ K. Then, comparing (7.5) and (8.3) we can define a multiplace representation as a
multi-G-module by the following formula
Π
(n,n′)
k


g1
...
gk

 (x) = ρ
(n)
k



g1
...
gk
x



. (8.9)
In a similar way, one can generalize to polyadic systems many other notions from group action
theory KIRILLOV [1976].

9. REGULAR MULTIACTIONS
The most important role in the study of polyadic systems is played by the case, when X =Gn,
and the multiaction coincides with the n-ary analog of translations MAL’TCEV [1954], so called
i-translations BELOUSOV [1972]. In the binary case, ordinary translations lead to regular representa-
tions KIRILLOV [1976], and therefore we call such an action a regular multiaction ρ
reg(n)
k . In
this connection, the analog of the Cayley theorem for n-ary groups was obtained in GAL’MAK [1986,
2001b]. Now we will show in examples, how the regular multiactions can arise from i-translations.
Example 9-1. Let G3 be a ternary semigroup, k = 2, and X =G3, then 2-place (left) action can be
defined as
ρ
reg(3)
2
g
h
u
def
= µ3 [g, h, u] . (9.1)
This gives the following composition law for two regular multiactions
ρ
reg(3)
2
g1
h1
ρ
reg(3)
2
g2
h2
u = µ3 [g1, h1, µ3 [g2, h2, u]]
= µ3 [µ3 [g1, h1, g2] , h2, u] = ρ
reg(3)
2
µ3 [g1, h1, g2]
h2
u .
(9.2)
Thus, using the regular 2-action (9.1) we have, in fact, derived the associative quiver corresponding
to (5.6).
The formula (9.1) can be simultaneously treated as a 2-translation BELOUSOV [1972]. In this way,
the following left regular multiaction
ρ
reg(n)
k



g1
...
gk
h



def
= µn [g1, . . . , gk, h] , (9.3)
corresponds to (5.10), where in the r.h.s. there is the i-translation with i = n. The right regular
multiaction corresponds to the i-translation with i = 1. The binary composition of the left regular
multiactions corresponds to (5.10). In general, the value of i fixes the minimal final arity n′
reg, which
differs for even and odd values of the initial arity n.
It follows from (9.3) that for regular multiactions the number of places is fixed
kreg = n − 1, (9.4)
and the arity changing formulas (5.17)–(5.18) become
n′
reg = n − ℓid (9.5)
n′
reg = ℓµ + 1. (9.6)
From (9.5)–(9.6) we conclude that for any n a regular multiaction having one multiplication ℓµ = 1
is binarizing and has n − 2 intact elements. For n = 3 see (9.2). Also, it follows from (9.5) that for
regular multiactions the number of intact elements gives exactly the difference between initial and
final arities.
If the initial arity is odd, then there exists a special middle regular multiaction generated by
the i-translation with i = (n + 1) 2. For n = 3 the corresponding associative quiver is (6.5) and
such 2-actions were used in BOROWIEC ET AL. [2006] to construct middle representations of ternary

32 STEVEN DUPLIJ
groups, which did not change arity (n′
= n). Here we give a more complicated example of a middle
regular multiaction, which can contain intact elements and can therefore change arity.
Example 9-2. Let us consider 5-ary semigroup and the following middle 4-action
ρ
reg(5)
4



g
h
u
v
s



= µ5

g, h,
i=3
↓
s , u, v

 . (9.7)
Using (9.6) we observe that there are two possibilities for the number of multiplications ℓµ = 2, 4.
The last case ℓµ = 4 is similar to the vertical associative quiver (6.5), but with a more complicated
l.h.s., that is
ρ
reg(5)
4



µ5 [g1, h1, g2, h2,g3]
µ5 [h3, g4, h4, g5, h5]
µ5 [u5, v5, u4, v4, u3]
µ5 [v3, u2, v2, u1, v1]
s



=
ρ
reg(5)
4



g1
h1
u1
v1
ρ
reg(5)
4



g2
h2
u2
v2
ρ
reg(5)
4



g3
h3
u3
v3
ρ
reg(5)
4



g4
h4
u4
v4
ρ
reg(5)
4



g5
h5
u5
v5
s















. (9.8)
Now we have an additional case with two intact elements ℓid and two multiplications ℓµ = 2 as
ρ
reg(5)
4



µ5 [g1, h1, g2, h2,g3]
h3
µ5 [h3, v3, u2, v2, u1]
v1
s



= ρ
reg(5)
4



g1
h1
u1
v1
ρ
reg(5)
4



g2
h2
u2
v2
ρ
reg(5)
4



g3
h3
u3
v3
s









,
(9.9)
with arity changing from n = 5 to n′
reg = 3. In addition to (9.9) we have 3 more possible regular
multiactions due to the associativity of µ5, when the multiplication brackets in the sequences of 6
elements in the first two rows and the second two ones can be shifted independently.
For n 3, in addition to left, right and middle multiactions, there exist intermediate cases. First,
observe that the i-translations with i = 2 and i = n − 1 immediately fix the final arity n′
reg = n.
Therefore, the composition of multiactions will be similar to (9.8), but with some permutations in the
l.h.s.
Now we consider some multiplace analogs of regular representations of binary groups KIRILLOV
[1976]. The straightforward generalization is to consider the previously introduced regular mul-
tiactions (9.3) in the r.h.s. of (8.9). Let Gn be a finite polyadic associative system and KGn
be a vector space spanned by Gn (some properties of n-ary group rings were considered in
ZEKOVI Ć AND ARTAMONOV [1999, 2002]). This means that any element of KGn can be uniquely
presented in the form w = l al · hl, al ∈ K, hl ∈ G. Then, using (9.3) and (8.9) we define the
i-regular k-place representation by
Π
reg(i)
k


g1
...
gk

 (w) =
l
al · µk+1 [g1 . . . gi−1hlgi+1 . . . gk] . (9.10)
Comparing (9.3) and (9.10) one can conclude that all the general properties of multiplace regular
representations are similar to those of the regular multiactions. If i = 1 or i = k, the multiplace

representation is called a right or left regular representation respectively. If k is even, the
representation with i = k 2 + 1 is called a middle regular representation. The case k = 2
was considered in BOROWIEC ET AL. [2006] for ternary groups.
10. MULTIPLACE REPRESENTATIONS OF TERNARY GROUPS
Let us consider the case n = 3, k = 2 in more detail, paying attention to its special peculiarities,
which corresponds to the 2-place (bi-element) representations of ternary groups BOROWIEC ET AL.
[2006]. Let V be a vector space over K and End V be a set of linear endomorphisms of V . From now
on we denote the ternary multiplication by square brackets only, as follows µ3 [g1, g2, g3] ≡ [g1g2g3],
and use the “horizontal” notation Π
g1
g2
≡ Π (g1, g2).
Definition 10-1. A left representation of a ternary group G, [ ]) in V is a map ΠL
: G × G →
End V such that
ΠL
(g1, g2) ◦ ΠL
(g3, g4) = ΠL
([g1g2g3] , g4) , (10.1)
ΠL
(g, g) = idV , (10.2)
where g, g1, g2, g3, g4 ∈ G.
Replacing in (10.2) g by g we obtain ΠL
(g, g) = idV , which means that in fact (10.2) has the form
ΠL
(g, g) = ΠL
(g, g) = idV , ∀g ∈ G. Note that the axioms considered in the above definition are
the natural ones satisfied by left multiplications g → [abg]. For all g1, g2, g3, g4 ∈ G we have
ΠL
([g1g2g3] , g4) = ΠL
(g1, [g2g3g4]) .
For all g, h, u ∈ G we have
ΠL
(g, h) = ΠL
([guu], h) = ΠL
(g, u) ◦ ΠL
(u, h) (10.3)
and
ΠL
(g, u) ◦ ΠL
(u, g) = ΠL
(u, g) ◦ ΠL
(g, u) = idV , (10.4)
and therefore every ΠL
(g, u) is invertible and ΠL
(g, u)
−1
= ΠL
(u, g). This means that any left
representation gives a representation of a ternary group by a binary group BOROWIEC ET AL. [2006].
If the ternary group is medial, then
ΠL
(g1, g2) ◦ ΠL
(g3, g4) = ΠL
(g3, g4) ◦ ΠL
(g1, g2) ,
i.e. the group so obtained is commutative. If the ternary group G, [ ] is commutative, then also
ΠL
(g, h) = ΠL
(h, g), because
ΠL
(g, h) = ΠL
(g, h)◦ΠL
(g, g) = ΠL
([g h g] , g) = ΠL
([h g g] , g) = ΠL
(h, g)◦ΠL
(g, g) = ΠL
(h, g) .
In the case of a commutative and idempotent ternary group any of its left representations is idempotent
and ΠL
(g, h)
−1
= ΠL
(g, h), so that commutative and idempotent ternary groups are represented
by Boolean groups.
Assertion 10-2. Let Let G, [ ] = der (G, ⊙) be a ternary group derived from a binary group G, ⊙ ,
then there is one-to-one correspondence between representations of (G, ⊙) and left representations
of (G, [ ]).

34 STEVEN DUPLIJ
Indeed, because (G, [ ]) = der (G, ⊙), then g ⊙h = [geh] and e = e, where e is unity of the binary
group (G, ⊙). If π ∈ Rep (G, ⊙), then (as it is not difficult to see) ΠL
(g, h) = π (g) ◦ π (h) is a left
representation of G, [ ] . Conversely, if ΠL
is a left representation of G, [ ] then π (g) = ΠL
(g, e)
is a representation of (G, ⊙). Moreover, in this case ΠL
(g, h) = π (g) ◦ π (h), because we have
ΠL
(g, h) = ΠL
(g, [ehe]) = ΠL
([geh], e) = ΠL
(g, e) ◦ ΠL
(h, e) = π (g) ◦ π (h) .
Let (G, [ ]) be a ternary group and (G × G, ∗) be a semigroup used in the construction of left
representations. According to Post POST [1940] one says that two pairs (a, b), (c, d) of elements of
G are equivalent, if there exists an element g ∈ G such that [abg] = [cdg]. Using a covering group we
can see that if this equation holds for some g ∈ G, then it holds also for all g ∈ G. This means that
ΠL
(a, b) = ΠL
(c, d) ⇐⇒ (a, b) ∼ (c, d),
i.e.
ΠL
(a, b) = ΠL
(c, d) ⇐⇒ [abg] = [cdg]
for some g ∈ G. Indeed, if [abg] = [cdg] holds for some g ∈ G, then
ΠL
(a, b) = ΠL
(a, b) ◦ ΠL
(g, g) = ΠL
([abg], g)
= ΠL
([cdg], g) = ΠL
(c, d) ◦ ΠL
(g, g) = ΠL
(c, d).
By analogy we can define
Definition 10-3. A right representation of a ternary group (G, [ ]) in V is a map ΠR
: G ×G →
End V such that
ΠR
(g3, g4) ◦ ΠR
(g1, g2) = ΠR
(g1, [g2g3g4]) , (10.5)
ΠR
(g, g) = idV , (10.6)
where g, g1, g2, g3, g4 ∈ G.
From (10.5)-(10.6) it follows that
ΠR
(g, h) = ΠR
(g, [u u h]) = ΠR
(u, h) ◦ ΠR
(g, u) . (10.7)
It is easy to check that ΠR
(g, h) = ΠL
h, g = ΠL
(g, h)
−1
. So it is sufficient to consider only
left representations (as in the binary case). Consider the following example of a group algebra ternary
generalization BOROWIEC ET AL. [2006].
Example 10-4. Let G be a ternary group and KG be a vector space spanned by G, which means that
any element of KG can be uniquely presented in the form t = n
i=1 kihi, ki ∈ K, hi ∈ G, n ∈ N (we
do not assume that G has finite rank). Then left and right regular representations are defined by
ΠL
reg (g1, g2) t =
n
i=1
ki [g1g2hi] , (10.8)
ΠR
reg (g1, g2) t =
n
i=1
ki [hig1g2] . (10.9)
Let us construct the middle representations as follows.
Definition 10-5. A middle representation of a ternary group G, [ ] in V is a map ΠM
: G×G →
End V such that
ΠM
(g3, h3) ◦ ΠM
(g2, h2) ◦ ΠM
(g1, h1) = ΠM
([g3g2g1] , [h1h2h3]) , (10.10)
ΠM
(g, h) ◦ ΠM
g, h = ΠM
g, h ◦ ΠM
(g, h) = idV (10.11)

It can be seen that a middle representation is a ternary group homomorphism ΠM
: G × Gop
→
der End V. Note that instead of (10.11) one can use ΠM
g, h ◦ ΠM
(g, h) = idV after changing g to
g and taking into account that g = g. In the case of idempotent elements g and h we have ΠM
(g, h) ◦
ΠM
(g, h) = idV , which means that the matrices ΠM
are Boolean. Thus all middle representation
matrices of idempotent ternary groups are Boolean. The composition ΠM
(g1, h1)◦ΠM
(g2, h2) is not
a middle representation, but the following proposition nevertheless holds.
Let ΠM
be a middle representation of a ternary group G, [ ] , then, if ΠL
u (g, h) = ΠM
(g, u) ◦
ΠM
(h, u) is a left representation of G, [ ] , then ΠL
u (g, h) ◦ ΠL
u′ (g′
, h′
) = ΠL
u′ ([ghu′
] , h′
), and,
if ΠR
u (g, h) = ΠM
(u, h)◦ΠM
(u, g) is a right representation of G, [ ] , then ΠR
u (g, h)◦ΠR
u′ (g′
, h′
) =
ΠR
u (g, [hg′
h′
]). In particular, ΠL
u (ΠR
u ) is a family of left (right) representations.
If a middle representation ΠM
of a ternary group G, [ ] satisfies ΠM
(g, g) = idV for all g ∈ G,
then it is a left and a right representation and ΠM
(g, h) = ΠM
(h, g) for all g, h ∈ G. Note that in
general ΠM
reg(g, g) = id. For regular representations we have the following commutation relations
ΠL
reg (g1, h1) ◦ ΠR
reg (g2, h2) = ΠR
reg (g2, h2) ◦ ΠL
reg (g1, h1) .
Let G, [ ] be a ternary group and let G × G, [ ]′
be a ternary group used in the construction of
the middle representation. In G, [ ] , and consequently in G × G, [ ]′
, we define the relation
(a, b) ∼ (c, d) ⇐⇒ [aub] = [cud]
for all u ∈ G. It is not difficult to see that this relation is a congruence in the ternary group
G × G, [ ]′
. For regular representations ΠM
reg(a, b) = ΠM
reg(c, d) if (a, b) ∼ (c, d). We have the
following relation
a a′
⇐⇒ a = [ga′
g] for some g ∈ G
or equivalently
a a′
⇐⇒ a′
= [gag] for some g ∈ G.
It is not difficult to see that it is an equivalence relation on G, [ ] , moreover, if G, [ ] is medial,
then this relation is a congruence.
Let G × G, [ ]′
be a ternary group used in a construction of middle representations, then
(a, b) (a′
, b) ⇐⇒ a′
= [gag] and
b′
= [hbb ] for some (g, h) ∈ G × G
is an equivalence relation on G × G, [ ]′
. Moreover, if (G, [ ]) is medial, then this relation is a
congruence. Unfortunately, however it is a weak relation. In a ternary group Z3, where [ghu] =
(g − h + u) (mod 3) we have only one class, i.e. all elements are equivalent. In Z4 with the operation
[ghu] = (g + h + u + 1) (mod 4) we have a a′
⇐⇒ a = a′
. However, for this relation the
following statement holds. If (a, b) (a′
, b′
), then
tr ΠM
(a, b) = tr ΠM
(a′
, b′
).
We have tr(AB) = tr(BA) for all A, B ∈ EndV , and
tr ΠM
(a, b) = tr ΠM
([ga′
g], [hb′
h] ) = tr ΠM
(g, h) ◦ ΠM
(a′
, b′
) ◦ ΠM
(g, h)
= tr ΠM
(g, h) ◦ ΠM
(g, h) ◦ ΠM
(a′
, b′
) = tr idV ◦ ΠM
(a′
b′
) = tr ΠM
(a′
, b′
)
In our derived case the connection with standard group representations is given by the following.
Let (G, ⊙) be a binary group, and the ternary derived group as G, [ ] = der (G, ⊙). There is
one-to-one correspondence between a pair of commuting binary group representations and a middle

36 STEVEN DUPLIJ
ternary derived group representation. Indeed, let π, ρ ∈ Rep (G, ⊙), π (g)◦ ρ (h) = ρ (h) ◦ π (g) and
ΠL
∈ Rep (G, [ ]). We take
ΠM
(g, h) = π (g) ◦ ρ h−1
, π (g) = ΠM
(g, e) , ρ (g) = ΠM
(e, g) .
Then using (10.10) we prove the needed representation laws.
Let G, [ ] be a fixed ternary group, G × G, [ ]′
a corresponding ternary group used in the con-
struction of middle representations, ((G × G)∗
, ⊛) a covering group of G × G, [ ]′
, (G × G, ⋄) =
ret(a,b)(G × G, ). If ΠM
(a, b) is a middle representation of G, [ ] , then π defined by
π(g, h, 0) = ΠM
(g, h), π(g, h, 1) = ΠM
(g, h) ◦ ΠM
(a, b)
is a representation of the covering group POST [1940]. Moreover
ρ(g, h) = ΠM
(g, h) ◦ ΠM
(a, b) = π(g, h, 1)
is a representation of the above retract induced by (a, b). Indeed, (a, b) is the identity of this retract
and ρ(a, b) = ΠM
(a, b) ◦ ΠM
(a, b) = idV . Similarly
ρ ((g, h) ⋄ (u, u)) = ρ ( (g, h), (a, b), (u, u) ) = ρ ([gau], [ubh]) = ΠM
([gau], [ubh])) ◦ ΠM
(a, b)
= ΠM
(g, h) ◦ ΠM
(a, b) ◦ ΠM
(u, u) ◦ ΠM
(a, b) = ρ(g, h) ◦ ρ(u, u)
But τ(g) = (g, g) is an embedding of (G, [ ]) into G × G, [ ]′
. Hence µ defined by µ(g, 0) =
ΠM
(g, g) and µ(g, 1) = ΠM
(g, g) ◦ ΠM
(a, a) is a representation of a covering group G∗
for (G, [ ])
(see the Post theorem POST [1940] for a = c). On the other hand, β(g) = ΠM
(g, g) ◦ ΠM
(a, a) is a
representation of a binary retract (G, · ) = reta(G, [ ]). Thus β can induce some middle representation
of (G, [ ]) (by the Gluskin-Hosszú theorem GLUSKIN [1965]).
Note that in the ternary group of quaternions K, [ ] (with norm 1), where [ghu] = ghu(−1) =
−ghu and gh is the multiplication of quaternions (−1 is a central element) we have 1 = −1, −1 = 1
and g = g for others. In K × K, [ ]′
we have (a, b) ∼ (−a, −b) and (a, −b) ∼ (−a, b), which gives
32 two-element equivalence classes. The embedding τ(g) = (g, g) suggest that ΠM
(i, i) = π(i) =
π(−i) = ΠM
(−i, −i). Generally ΠM
(a, b) = ΠM
(−a, −b) and ΠM
(a, −b) = ΠM
(−a, b).
The relation (a, b) ∼ (c, d) ⇐⇒ [abg] = [cdg] for all g ∈ G is a congruence on (G × G, ∗). Note
that this relation can be defined as ”for some g”. Indeed, using a covering group we can see that if
[abg] = [cdg] holds for some g then it holds also for all g. Thus πL
(a, b) = ΠL
(c, d) ⇐⇒ (a, b) ∼
(c, d). Indeed
ΠL
(a, b) = ΠL
(a, b) ◦ ΠL
(g, g) = ΠL
([a b g], g)
= ΠL
([c d g], g) = ΠL
(c, d) ◦ ΠL
(g, g) = ΠL
(c, d).
We conclude, that every left representation of a commutative group G, [ ] is a middle representa-
tion. Indeed,
ΠL
(g, h) ◦ ΠL
(g, h) = ΠL
([g h g], h) = ΠL
([g g h], h) = ΠL
(h, h) = idV
and
ΠL
(g1, g2) ◦ ΠL
(g3, g4) ◦ ΠL
(g5, g6) = ΠL
([[g1g2g3]g4g5], g6) = ΠL
([[g1g3g2]g4g5], g6)
= ΠL
([g1g3[g2g4g5]], g6) = ΠL
([g1g3[g5g4g2]], g6) = ΠL
([g1g3g5], [g4g2g6]) = ΠL
([g1g3g5], [g6g4g2]).
Note that the converse holds only for the special kind of middle representations such that
ΠM
(g, g) = idV . Therefore,

Assertion 10-6. There is one-one correspondence between left representations of G, [ ] and binary
representations of the retract reta(G, [ ]).
Indeed, let ΠL
(g, a) be given, then ρ(g) = ΠL
(g, a) is such representation of the retract, as can be
directly shown. Conversely, assume that ρ(g) is a representation of the retract reta(G, [ ]). Deﬁne
ΠL
(g, h) = ρ(g) ◦ ρ(h)−1
, then ΠL
(g, h) ◦ ΠL
(u, u) = ρ(g) ◦ ρ(h)−1
◦ ρ(u) ◦ ρ(u)−1
= ρ(g ⊛(h)−1
◦
⊛u) ◦ ρ(u)−1
= ρ([ [ g a [ a h a ] ] a u ]) ◦ ρ(u)−1
= ρ([ g h g ]) ◦ ρ(u)−1
= ΠL
([ g h u ], u),
11. MATRIX REPRESENTATIONS OF TERNARY GROUPS
Here we give several examples of matrix representations for concrete ternary groups. Let G =
Z3 ∋ {0, 1, 2} and the ternary multiplication be [ghu] = g − h + u. Then [ghu] = [uhg] and 0 = 0,
1 = 1, 2 = 2, therefore (G, [ ]) is an idempotent medial ternary group. Thus ΠL
(g, h) = ΠR
(h, g)
and
ΠL
(a, b) = ΠL
(c, d) ⇐⇒ (a − b) = (c − d)mod 3. (11.1)
The calculations give the left regular representation in the manifest matrix form
ΠL
reg (0, 0) = ΠL
reg (2, 2) = ΠL
reg (1, 1) = ΠR
reg (0, 0)
= ΠR
reg (2, 2) = ΠR
reg (1, 1) =


1 0 0
0 1 0
0 0 1

 = [1] ⊕ [1] ⊕ [1], (11.2)
ΠL
reg (2, 0) = ΠL
reg (1, 2) = ΠL
reg (0, 1) = ΠR
reg (2, 1) = ΠR
reg (1, 0) = ΠR
reg (0, 2) =


0 1 0
0 0 1
1 0 0


= [1] ⊕



−
1
2
−
√
3
2√
3
2
−
1
2


 = [1] ⊕ −
1
2
+
1
2
i
√
3 ⊕ −
1
2
−
1
2
i
√
3 , (11.3)
ΠL
reg (2, 1) = ΠL
reg (1, 0) = ΠL
reg (0, 2) = ΠR
reg (2, 0) = ΠR
reg (1, 2) = ΠR
reg (0, 1) =


0 0 1
1 0 0
0 1 0


= [1] ⊕



−
1
2
√
3
2
−
√
3
2
−
1
2


 = [1] ⊕ −
1
2
−
1
2
i
√
3 ⊕ −
1
2
+
1
2
i
√
3 . (11.4)
Consider next the middle representation construction. The middle regular representation is deﬁned
by
ΠM
reg (g1, g2) t =
n
i=1
ki [g1hig2] .
For regular representations we have
ΠM
reg (g1, h1) ◦ ΠR
reg (g2, h2) = ΠR
reg (h2, h1) ◦ ΠM
reg (g1, g2) , (11.5)
ΠM
reg (g1, h1) ◦ ΠL
reg (g2, h2) = ΠL
reg (g1, g2) ◦ ΠM
reg (h2, h1) . (11.6)

38 STEVEN DUPLIJ
For the middle regular representation matrices we obtain
ΠM
reg (0, 0) = ΠM
reg (1, 2) = ΠM
reg (2, 1) =


1 0 0
0 0 1
0 1 0

 ,
ΠM
reg (0, 1) = ΠM
reg (1, 0) = ΠM
reg (2, 2) =


0 1 0
1 0 0
0 0 1

 ,
ΠM
reg (0, 2) = ΠM
reg (2, 0) = ΠM
reg (1, 1) =


0 0 1
0 1 0
1 0 0

 .
The above representation ΠM
reg of Z3, [ ] is equivalent to the orthogonal direct sum of two irre-
ducible representations
ΠM
reg (0, 0) = ΠM
reg (1, 2) = ΠM
reg (2, 1) = [1] ⊕
−1 0
0 1
,
ΠM
reg (0, 1) = ΠM
reg (1, 0) = ΠM
reg (2, 2) = [1] ⊕



1
2
−
√
3
2
−
√
3
2
−
1
2


 ,
ΠM
reg (0, 2) = ΠM
reg (2, 0) = ΠM
reg (1, 1) = [1] ⊕



1
2
√
3
2√
3
2
−
1
2


 ,
i.e. one-dimensional trivial [1] and two-dimensional irreducible. Note, that in this example
ΠM
(g, g) = ΠM
(g, g) = idV , but ΠM
(g, h) ◦ ΠM
(g, h) = idV , and so ΠM
are of the second degree.
Consider a more complicated example of left representations. Let G = Z4 ∋ {0, 1, 2, 3} and the
ternary multiplication be
[ghu] = (g + h + u + 1) mod 4. (11.7)
We have the multiplication table
[g, h, 0] =




1 2 3 0
2 3 0 1
3 0 1 2
0 1 2 3



 [g, h, 1] =




2 3 0 1
3 0 1 2
0 1 2 3
1 2 3 0




[g, h, 2] =




3 0 1 2
0 1 2 3
1 2 3 0
2 3 0 1



 [g, h, 3] =




0 1 2 3
1 2 3 0
2 3 0 1
3 0 1 2




Then the skew elements are 0 = 3, 1 = 2, 2 = 1, 3 = 0, and therefore (G, [ ]) is a (non-idempotent)
commutative ternary group. The left representation is deﬁned by the expansion ΠL
reg (g1, g2) t =
n
i=1 ki [g1g2hi], which means that (see the general formula (9.10))
ΠL
reg (g, h) |u = | [ghu] .

Analogously, for right and middle representations
ΠR
reg (g, h) |u = | [ugh] , ΠM
reg (g, h) |u = | [guh] .
Therefore | [ghu] = | [ugh] = | [guh] and
ΠL
reg (g, h) = ΠR
reg (g, h) |u = ΠM
reg (g, h) |u ,
so ΠL
reg (g, h) = ΠR
reg (g, h) = ΠM
reg (g, h). Thus it is sufﬁcient to consider the left representation only.
In this case the equivalence is ΠL
(a, b) = ΠL
(c, d) ⇐⇒ (a + b) = (c + d)mod 4, and we obtain
the following classes
ΠL
reg (0, 0) = ΠL
reg (1, 3) = ΠL
reg (2, 2) = ΠL
reg (3, 1) =




0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0



 = [1] ⊕ [−1] ⊕ [−i] ⊕ [i] ,
ΠL
reg (0, 1) = ΠL
reg (1, 0) = ΠL
reg (2, 3) = ΠL
reg (3, 2) =




0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0



 = [1] ⊕ [−1] ⊕ [−1] ⊕ [−1] ,
ΠL
reg (0, 2) = ΠL
reg (1, 1) = ΠL
reg (2, 0) = ΠL
reg (3, 3) =




0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0



 = [1] ⊕ [−1] ⊕ [i] ⊕ [−i] ,
ΠL
reg (0, 3) = ΠL
reg (1, 2) = ΠL
reg (2, 1) = ΠL
reg (3, 0) =




1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1



 = [1] ⊕ [−1] ⊕ [1] ⊕ [1] .
It is seen that, due to the fact that the ternary operation (11.7) is commutative, there are only one-
dimensional irreducible left representations.
Let us “algebralize” the above regular representations in the following way. From (10.1) we have,
for the left representation
ΠL
reg (i, j) ◦ ΠL
reg (k, l) = ΠL
reg (i, [jkl]) , (11.8)
where [jkl] = j − k + l, i, j, k, l ∈ Z3. Denote γL
i = ΠL
reg (0, i), i ∈ Z3, then we obtain the algebra
with the relations
γL
i γL
j = γL
i+j. (11.9)
Conversely, any matrix representation of γiγj = γi+j leads to the left representation by ΠL
(i, j) =
γj−i. In the case of the middle regular representation we introduce γM
k+l = ΠM
reg (k, l), k, l ∈ Z3, then
we obtain
γM
i γM
j γM
k = γM
[ijk], i, j, k ∈ Z3. (11.10)
In some sense (11.10) can be treated as a ternary analog of the Clifford algebra. As
before, any matrix representation of (11.10) gives the middle representation ΠM
(k, l) = γk+l.

40 STEVEN DUPLIJ
12. TERNARY ALGEBRAS AND HOPF ALGEBRAS
Let us consider associative ternary algebras CARLSSON [1980], DE AZCARRAGA AND IZQUIERDO
[2010]. One can introduce an autodistributivity property [[xyz] ab] = [[xab] [yab] [zab]] (see DUDEK
[1993]). If we take 2 ternary operations { , , } and [ , , ], then distributivity is given by {[xyz] ab} =
[{xab} {yab} {zab}]. If (+) is a binary operation (addition), then left linearity is
[(x + z) , a, b] = [xab] + [zab] . (12.1)
By analogy one can define central (middle) and right linearity. Linearity is defined, when left,
middle and right linearity hold simultaneously.
Definition 12-1. An associative ternary algebra is a triple A, µ3, η(3)
, where A is a lin-
ear space over a field K, µ3 is a linear map A ⊗ A ⊗ A → A called ternary multiplication
µ3 (a ⊗ b ⊗ c) = [abc] which is ternary associative [[abc] de] = [a [bcd] e] = [ab [cde]] or
µ3 ◦ (µ3 ⊗ id ⊗ id) = µ3 ◦ (id ⊗µ3 ⊗ id) = µ3 ◦ (id ⊗ id ⊗µ3) . (12.2)
There are two types DUPLIJ [2001] of ternary unit maps η(3)
: K → A:
1) One strong unit map
µ3 ◦ η(3)
⊗ η(3)
⊗ id = µ3 ◦ η(3)
⊗ id ⊗η(3)
= µ3 ◦ id ⊗η(3)
⊗ η(3)
= id; (12.3)
2) Two sequential units η
(3)
1 and η
(3)
2 satisfying
µ3 ◦ η
(3)
1 ⊗ η
(3)
2 ⊗ id = µ3 ◦ η
(3)
1 ⊗ id ⊗η
(3)
2 = µ3 ◦ id ⊗η
(3)
1 ⊗ η
(3)
2 = id; (12.4)
In first case the ternary analog of the binary relation η(2)
(x) = x1, where x ∈ K, 1 ∈ A, is
η(3)
(x) = [x, 1, 1] = [1, 1, x] = [1, x, 1] . (12.5)
Let (A, µA, ηA), (B, µB, ηB) and (C, µC, ηC) be ternary algebras, then the ternary tensor
product space A ⊗ B ⊗ C is naturally endowed with the structure of an algebra. The multipli-
cation µA⊗B⊗C on A ⊗ B ⊗ C reads
[(a1 ⊗ b1 ⊗ c1)(a2 ⊗ b2 ⊗ c2)(a3 ⊗ b3 ⊗ c3)] = [a1a2a3] ⊗ [b1b2b3] ⊗ [c1c2c3] , (12.6)
and so the set of ternary algebras is closed under taking ternary tensor products. A ternary algebra
map (homomorphism) is a linear map between ternary algebras f : A → B which respects the ternary
algebra structure
f ([xyz]) = [f (x) , f (y) , f (z)] , (12.7)
f (1A) = 1B. (12.8)
Let C be a linear space over a field K.
Definition 12-2. A ternary comultiplication ∆(3)
is a linear map over a field K such that
∆3 : C → C ⊗ C ⊗ C. (12.9)
In the standard Sweedler notations SWEEDLER [1969] ∆3 (a) = n
i=1 a′
i ⊗a′′
i ⊗a′′′
i = a(1) ⊗a(2) ⊗
a(3). Consider different possible types of ternary coassociativity DUPLIJ [2001], BOROWIEC ET AL.
[2001].
(1) A standard ternary coassociativity
(∆3 ⊗ id ⊗ id) ◦ ∆3 = (id ⊗∆3 ⊗ id) ◦ ∆3 = (id ⊗ id ⊗∆3) ◦ ∆3, (12.10)

(2) A nonstandard ternary Σ-coassociativity (Gluskin-type positional operatives)
(∆3 ⊗ id ⊗ id) ◦ ∆3 = (id ⊗ (σ ◦ ∆3) ⊗ id) ◦ ∆3,
where σ ◦ ∆3 (a) = ∆3 (a) = a(σ(1)) ⊗ a(σ(2)) ⊗ a(σ(3)) and σ ∈ Σ ⊂ S3.
(3) A permutational ternary coassociativity
(∆3 ⊗ id ⊗ id) ◦ ∆3 = π ◦ (id ⊗∆3 ⊗ id) ◦ ∆3,
where π ∈ Π ⊂ S5.
A ternary comediality is
(∆3 ⊗ ∆3 ⊗ ∆3) ◦ ∆3 = σmedial ◦ (∆3 ⊗ ∆3 ⊗ ∆3) ◦ ∆3,
where σmedial = 123456789
147258369
∈ S9. A ternary counit is defined as a map ε(3)
: C → K. In general,
ε(3)
= ε(2)
satisfying one of the conditions below. If ∆3 is derived, then maybe ε(3)
= ε(2)
, but another
counits may exist. There are two types of ternary counits:
(1) Standard (strong) ternary counit
(ε(3)
⊗ ε(3)
⊗ id) ◦ ∆3 = (ε(3)
⊗ id ⊗ε(3)
) ◦ ∆3 = (id ⊗ε(3)
⊗ ε(3)
) ◦ ∆3 = id, (12.11)
(2) Two sequential (polyadic) counits ε
(3)
1 and ε
(3)
2
(ε
(3)
1 ⊗ ε
(3)
2 ⊗ id) ◦ ∆ = (ε
(3)
1 ⊗ id ⊗ε
(3)
2 ) ◦ ∆ = (id ⊗ε
(3)
1 ⊗ ε
(3)
2 ) ◦ ∆ = id, (12.12)
Below we will consider only the first standard type of associativity (12.10). The σ-
cocommutativity is defined as σ ◦ ∆3 = ∆3.
Definition 12-3. A ternary coalgebra is a triple C, ∆3, ε(3)
, where C is a linear space and ∆3
is a ternary comultiplication (12.9) which is coassociative in one of the above senses and ε(3)
is one
of the above counits.
Let A, µ(3)
be a ternary algebra and (C, ∆3) be a ternary coalgebra and f, g, h ∈ HomK (C, A).
Ternary convolution product is
[f, g, h]∗ = µ(3)
◦ (f ⊗ g ⊗ h) ◦ ∆3 (12.13)
or in the Sweedler notation [f, g, h]∗ (a) = f a(1) g a(2) h a(3) .
Definition 12-4. A ternary coalgebra is called derived, if there exists a binary (usual, see e.g.
SWEEDLER [1969]) coalgebra ∆2 : C → C ⊗ C such that
∆3,der = (id ⊗∆2) ⊗ ∆2. (12.14)
Definition 12-5. A ternary bialgebra B is B, µ(3)
, η(3)
, ∆3, ε(3)
for which B, µ(3)
, η(3)
is a ternary
algebra and B, ∆3, ε(3)
is a ternary coalgebra and they are compatible
∆3 ◦ µ(3)
= µ(3)
◦ ∆3 (12.15)
One can distinguish four kinds of ternary bialgebras with respect to a “being derived” property:
(1) A ∆-derived ternary bialgebra
∆3 = ∆3,der = (id ⊗∆2) ◦ ∆2 (12.16)
(2) A µ-derived ternary bialgebra
µ
(3)
der = µ
(3)
der = µ(2)
◦ µ(2)
⊗ id (12.17)
(3) A derived ternary bialgebra is simultaneously µ-derived and ∆-derived ternary bialgebra.

42 STEVEN DUPLIJ
(4) A non-derived ternary bialgebra which does not satisfy (12.16) and (12.17).
Possible types of ternary antipodes can be defined by analogy with binary coalgebras.
Definition 12-6. A skew ternary antipod is
µ(3)
◦(S
(3)
skew⊗id ⊗ id)◦∆3 = µ(3)
◦(id ⊗S
(3)
skew⊗id)◦∆3 = µ(3)
◦(id ⊗ id ⊗S
(3)
skew)◦∆3 = id . (12.18)
If only one equality from (12.18) is satisfied, the corresponding skew antipode is called left,
middle or right.
Definition 12-7. Strong ternary antipode is
µ(2)
⊗ id ◦ (id ⊗S
(3)
strong ⊗ id) ◦ ∆3 = 1 ⊗ id, id ⊗µ(2)
◦ (id ⊗ id ⊗S
(3)
strong) ◦ ∆3 = id ⊗1,
where 1 is a unit of algebra.
If in a ternary coalgebra the relation
∆3 ◦ S = τ13 ◦ (S ⊗ S ⊗ S) ◦ ∆3 (12.19)
holds true, where τ13 = 123
321
, then it is called skew-involutive.
Definition 12-8. A ternary Hopf algebra H, µ(3)
, η(3)
, ∆3, ε(3)
, S(3)
is a ternary bialgebra with a
ternary antipode S(3)
of the corresponding above type .
Let us consider concrete constructions of ternary comultiplications, bialgebras and Hopf algebras.
A ternary group-like element can be defined by ∆3 (g) = g ⊗ g ⊗ g, and for 3 such elements
we have
∆3 ([g1g2g3]) = ∆3 (g1) ∆3 (g2) ∆3 (g3) . (12.20)
But an analog of the binary primitive element (satisfying ∆(2)
(x) = x ⊗ 1 + 1 ⊗ x) cannot be chosen
simply as ∆3 (x) = x ⊗ e ⊗ e + e ⊗ x ⊗ e + e ⊗ e ⊗ x, since the algebra structure is not preserved.
Nevertheless, if we introduce two idempotent units e1, e2 satisfying “semiorthogonality” [e1e1e2] = 0,
[e2e2e1] = 0, then
∆3 (x) = x ⊗ e1 ⊗ e2 + e2 ⊗ x ⊗ e1 + e1 ⊗ e2 ⊗ x (12.21)
and now ∆3 ([x1x2x3]) = [∆3 (x1) ∆3 (x2) ∆3 (x3)]. Using (12.21) ε (x) = 0, ε (e1,2) = 1, and
S(3)
(x) = −x, S(3)
(e1,2) = e1,2, one can construct a ternary universal enveloping algebra in full
analogy with the binary case (see e.g. KASSEL [1995]).
One of the most important examples of noncocommutative Hopf algebras is the well known
Sweedler Hopf algebra SWEEDLER [1969] which in the binary case has two generators x and y
satisfying
µ(2)
(x, x) = 1, (12.22)
µ(2)
(y, y) = 0, (12.23)
σ
(2)
+ (xy) = −σ
(2)
− (xy) . (12.24)
It has the following comultiplication
∆2 (x) = x ⊗ x, (12.25)
∆2 (y) = y ⊗ x + 1 ⊗ y, (12.26)

counit ε(2)
(x) = 1, ε(2)
(y) = 0, and antipod S(2)
(x) = x, S(2)
(y) = −y, which respect the algebra
structure. In the derived case a ternary Sweedler algebra is generated also by two generators x
and y obeying DUPLIJ [2001]
µ(3)
(x, e, x) = µ(3)
(e, x, x) = µ(3)
(x, x, e) = e, (12.27)
σ
(3)
+ ([yey]) = 0, (12.28)
σ
(3)
+ ([xey]) = −σ
(3)
− ([xey]) . (12.29)
The derived Hopf algebra structure is given by
∆3 (x) = x ⊗ x ⊗ x, (12.30)
∆3 (y) = y ⊗ x ⊗ x + e ⊗ y ⊗ x + e ⊗ e ⊗ y, (12.31)
ε(3)
(x) = ε(2)
(x) = 1, (12.32)
ε(3)
(y) = ε(2)
(y) = 0, (12.33)
S(3)
(x) = S(2)
(x) = x, (12.34)
S(3)
(y) = S(2)
(y) = −y, (12.35)
and it can be checked that (12.30)-(12.32) are algebra maps, while (12.34) are antialgebra maps. To
obtain a non-derived ternary Sweedler example we have the following possibilities: 1) one “even”
generator x, two “odd” generators y1,2 and one ternary unit e; 2) two “even” generators x1,2, one
“odd” generator y and two ternary units e1,2. In the ﬁrst case the ternary algebra structure is (no
summation, i = 1, 2)
[xxx] = e, (12.36)
[yiyiyi] = 0, (12.37)
σ
(3)
+ ([yixyi]) = σ
(3)
+ ([xyix]) = 0, (12.38)
[xeyi] = − [xyie] ,
[exyi] = − [yixe] , (12.39)
[eyix] = − [yiex] , (12.40)
σ
(3)
+ ([y1xy2]) = −σ
(3)
− ([y1xy2]) . (12.41)
The corresponding ternary Hopf algebra structure is
∆3 (x) = x ⊗ x ⊗ x, ∆3 (y1,2) = y1,2 ⊗ x ⊗ x + e1,2 ⊗ y2,1 ⊗ x + e1,2 ⊗ e2,1 ⊗ y2,1, (12.42)
ε(3)
(x) = 1, ε(3)
(yi) = 0, (12.43)
S(3)
(x) = x, S(3)
(yi) = −yi. (12.44)
In the second case we have for the algebra structure
[xixjxk] = δijδikδjkei, [yyy] = 0, (12.45)
σ
(3)
+ ([yxiy]) = 0, σ
(3)
+ ([xiyxi]) = 0, (12.46)
σ
(3)
+ ([y1xy2]) = 0, σ
(3)
− ([y1xy2]) = 0, (12.47)

44 STEVEN DUPLIJ
and the ternary Hopf algebra structure is DUPLIJ [2001]
∆3 (xi) = xi ⊗ xi ⊗ xi,
∆3 (y) = y ⊗ x1 ⊗ x1 + e1 ⊗ y ⊗ x2 + e1 ⊗ e2 ⊗ y, (12.48)
ε(3)
(xi) = 1, (12.49)
ε(3)
(y) = 0, (12.50)
S(3)
(xi) = xi, (12.51)
S(3)
(y) = −y. (12.52)
13. TERNARY QUANTUM GROUPS
A ternary commutator can be obtained in different ways BREMNER AND HENTZEL [2000].
We will consider the simplest version called a Nambu bracket (see e.g. TAKHTAJAN [1994],
DE AZCARRAGA AND IZQUIERDO [2010]). Let us introduce two maps ω
(3)
± : A⊗A⊗A → A⊗A⊗A
by
ω
(3)
+ (a ⊗ b ⊗ c) = a ⊗ b ⊗ c + b ⊗ c ⊗ a + c ⊗ a ⊗ b, (13.1)
ω
(3)
− (a ⊗ b ⊗ c) = b ⊗ a ⊗ c + c ⊗ b ⊗ a + a ⊗ c ⊗ b. (13.2)
Thus, obviously µ(3)
◦ ω
(3)
± = σ
(3)
± ◦ µ(3)
, where σ
(3)
± ∈ S3 denotes a sum of terms having even and
odd permutations respectively. In the binary case ω
(2)
+ = id ⊗ id and ω
(2)
− = τ is the twist operator
τ : a ⊗ b → b ⊗ a, while µ(2)
◦ ω
(2)
− is permutation σ
(2)
− (ab) = ba. So the Nambu product is
ω
(3)
N = ω
(3)
+ − ω
(3)
− , and the ternary commutator is [, , ]N = σ
(3)
N = σ
(3)
+ − σ
(3)
− , or TAKHTAJAN [1994]
[a, b, c]N = [abc] + [bca] + [cab] − [cba] − [acb] − [bac] (13.3)
An abelian ternary algebra is deﬁned by the vanishing of the Nambu bracket [a, b, c]N = 0 or
ternary commutation relation σ
(3)
+ = σ
(3)
− . By analogy with the binary case a deformed ternary
algebra can be deﬁned by
σ
(3)
+ = qσ
(3)
− or [abc] + [bca] + [cab] = q ([cba] + [acb] + [bac]) , (13.4)
where multiplication by q is treated as an external operation.
Let us consider a ternary analog of the Woronowicz example of a bialgebra construction, which in
the binary case has two generators satisfying xy = qyx (or σ
(2)
+ (xy) = qσ
(2)
− (xy)), then the following
coproducts
∆2 (x) = x ⊗ x (13.5)
∆2 (y) = y ⊗ x + 1 ⊗ y (13.6)
are algebra maps. In the derived ternary case using (13.4) we have
σ
(3)
+ ([xey]) = qσ
(3)
− ([xey]) , (13.7)
where e is the ternary unit and ternary coproducts are
∆3 (e) = e ⊗ e ⊗ e, (13.8)
∆3 (x) = x ⊗ x ⊗ x, (13.9)
∆3 (y) = y ⊗ x ⊗ x + e ⊗ y ⊗ x + e ⊗ e ⊗ y, (13.10)

which are ternary algebra maps, i.e. they satisfy
σ
(3)
+ ([∆3 (x) ∆3 (e) ∆3 (y)]) = qσ
(3)
− ([∆3 (x) ∆3 (e) ∆3 (y)]) . (13.11)
Let us consider the group G = SL (n, K). Then the algebra generated by ai
j ∈ SL (n, K) can be
endowed with the structure of a ternary Hopf algebra (see, e.g., MADORE [1995] for the binary case)
by choosing the ternary coproduct, counit and antipode as (here summation is implied)
∆3 ai
j = ai
k ⊗ ak
l ⊗ al
j, ε ai
j = δi
j, S(3)
ai
j = a−1 i
j
. (13.12)
This antipode is a skew one since from (12.18) it follows that
µ(3)
◦ (S(3)
⊗ id ⊗ id) ◦ ∆3 ai
j = S(3)
ai
k ak
l al
j = a−1 i
k
ak
l al
j = δi
l al
j = ai
j. (13.13)
This ternary Hopf algebra is derived since for ∆(2)
= ai
j ⊗ aj
k we have
∆3 = id ⊗∆(2)
⊗ ∆(2)
ai
j = id ⊗∆(2)
ai
k ⊗ ak
j = ai
k ⊗ ∆(2)
ak
j = ai
k ⊗ ak
l ⊗ al
j. (13.14)
In the most important case n = 2 we can obtain the manifest action of the ternary coproduct ∆3 on
components. Possible non-derived matrix representations of the ternary product can be done only by
four-rank n × n × n × n twice covariant and twice contravariant tensors aij
kl . Among all products
the non-derived ones are only the following: aoi
jkbjl
oocko
il and aij
okbol
iocko
il (where o is any index). So using
e.g. the first choice we can define the non-derived Hopf algebra structure by
∆3 aij
kl = aiµ
vρ ⊗ avσ
kl ⊗ aρj
µσ, (13.15)
ε aij
kl =
1
2
δi
kδj
l + δi
l δj
k , (13.16)
and the skew antipod sij
kl = S(3)
aij
kl which is a solution of the equation siµ
vρavσ
kl = δi
ρδµ
k δσ
l .
Next consider ternary dual pair k (G) (push-forward) and F (G) (pull-back) which are related by
k∗
(G) ∼= F (G) (see e.g. KOGORODSKI AND SOIBELMAN [1998]). Here k (G) = span (G) is a
ternary group algebra (G has a ternary product [ ]G or µ
(3)
G ) over a field k. If u ∈ k (G) (u = ui
xi, xi ∈
G), then
[uvw]k = ui
vj
wl
[xixjxl]G (13.17)
is associative, and so (k (G) , [ ]k) becomes a ternary algebra. Define a ternary coproduct ∆3 :
k (G) → k (G) ⊗ k (G) ⊗ k (G) by
∆3 (u) = ui
xi ⊗ xi ⊗ xi (13.18)
(derived and associative), then ∆3 ([uvw]k) = [∆3 (u) ∆3 (v) ∆3 (w)]k, and k (G) is a ternary bial-
gebra. If we define a ternary antipod by S
(3)
k = ui
¯xi, where ¯xi is a skew element of xi, then k (G)
becomes a ternary Hopf algebra.
In the dual case of functions F (G) : {ϕ : G → k} a ternary product [ ]F or µ
(3)
F (derived and
associative) acts on ψ (x, y, z) as
µ
(3)
F ψ (x) = ψ (x, x, x) , (13.19)
and so F (G) is a ternary algebra. Let F (G) ⊗ F (G) ⊗ F (G) ∼= F (G × G × G), then we define a
ternary coproduct ∆3 : F (G) → F (G) ⊗ F (G) ⊗ F (G) as
(∆3ϕ) (x, y, z) = ϕ ([xyz]F ) , (13.20)

46 STEVEN DUPLIJ
which is derive and associative. Thus we can obtain ∆3 ([ϕ1ϕ2ϕ3]F ) = [∆3 (ϕ1) ∆3 (ϕ2) ∆3 (ϕ3)]F ,
and therefore F (G) is a ternary bialgebra. If we define a ternary antipod by
S
(3)
F (ϕ) = ϕ (¯x) , (13.21)
where ¯x is a skew element of x, then F (G) becomes a ternary Hopf algebra.
Let us introduce a ternary analog of the R-matrix DUPLIJ [2001]. For a ternary Hopf algebra H
we consider a linear map R(3)
: H ⊗ H ⊗ H → H ⊗ H ⊗ H.
Definition 13-1. A ternary Hopf algebra H, µ(3)
, η(3)
, ∆3, ε(3)
, S(3)
is called quasifiveangular4
if it satisfies
(∆3 ⊗ id ⊗ id) = R
(3)
145R
(3)
245R
(3)
345, (13.22)
(id ⊗∆3 ⊗ id) = R
(3)
125R
(3)
145R
(3)
135, (13.23)
(id ⊗ id ⊗∆3) = R
(3)
125R
(3)
124R
(3)
123, (13.24)
where as usual the index of R denotes action component positions.
Using the standard procedure (see, e.g., KASSEL [1995], CHARI AND PRESSLEY [1996], MAJID
[1995]) we obtain a set of abstract ternary quantum Yang-Baxter equations, one of which has
the form DUPLIJ [2001]
R
(3)
243R
(3)
342R
(3)
125R
(3)
145R
(3)
135 = R
(3)
123R
(3)
132R
(3)
145R
(3)
245R
(3)
345, (13.25)
and others can be obtained by corresponding permutations. The classical ternary Yang-Baxter equa-
tions form a one parameter family of solutions R (t) can be obtained by the expansion
R(3)
(t) = e ⊗ e ⊗ e + rt + O t2
, (13.26)
where r is a ternary classical R-matrix, then e.g. for (13.25) we have
r342r125r145r135 + r243r125r145r135 + r243r342r145r135 + r243r342r125r135 + r243r342r125r145
= r132r145r245r345 + r123r145r245r345 + r123r132r245r345 + r123r132r145r345 + r123r132r145r245.
For three ternary Hopf algebras HA,B,C, µ
(3)
A,B,C, η
(3)
A,B,C, ∆
(3)
A,B,C, ε
(3)
A,B,C, S
(3)
A,B,C we can introduce
a non-degenerate ternary “pairing” (see, e.g., CHARI AND PRESSLEY [1996] for the binary case)
, , (3)
: HA × HB × HC → K, trilinear over K, satisfying DUPLIJ [2001]
η
(3)
A (a) , b, c
(3)
= a, ε
(3)
B (b) , c
(3)
, a, η
(3)
B (b) , c
(3)
= ε
(3)
A (a) , b, c
(3)
,
b, η
(3)
B (b) , c
(3)
= a, b, ε
(3)
C (c)
(3)
, a, b, η
(3)
C (c)
(3)
= a, ε
(3)
B (b) , c
(3)
,
a, b, η
(3)
C (c)
(3)
= ε
(3)
A (a) , b, c
(3)
, η
(3)
A (a) , b, c
(3)
= a, b, ε
(3)
C (c)
(3)
,
4
The reason for such notation is clear from (13.25).

µ
(3)
A (a1 ⊗ a2 ⊗ a3) , b, c
(3)
= a1 ⊗ a2 ⊗ a3, ∆
(3)
B (b) , c
(3)
,
∆
(3)
A (a) , b1 ⊗ b2 ⊗ b3, c
(3)
= a, µ
(3)
B (b1 ⊗ b2 ⊗ b3) , c
(3)
,
a, µ
(3)
B (b1 ⊗ b2 ⊗ b3) , c
(3)
= a, b1 ⊗ b2 ⊗ b3, ∆
(3)
C (c)
(3)
,
a, ∆
(3)
B (b) , c1 ⊗ c2 ⊗ c3
(3)
= a, b, µ
(3)
C (c1 ⊗ c2 ⊗ c3)
(3)
,
a, b, µ
(3)
C (c1 ⊗ c2 ⊗ c3)
(3)
= ∆
(3)
A (a) , b, c1 ⊗ c2 ⊗ c3
(3)
,
a1 ⊗ a2 ⊗ a3, b, ∆
(3)
C (c)
(3)
= µ
(3)
A (a1 ⊗ a2 ⊗ a3) , b, c
(3)
,
S
(3)
A (a) , b, c
(3)
= a, S
(3)
B (b) , c
(3)
= a, b, S
(3)
C (c)
(3)
,
where a, ai ∈ HA, b, bi ∈ HB. The ternary “paring” between HA ⊗ HA ⊗ HA and HB ⊗ HB ⊗ HB
is given by a1 ⊗ a2 ⊗ a3, b1 ⊗ b2 ⊗ b3
(3)
= a1, b1
(3)
a2, b2
(3)
a3, b3
(3)
. These constructions can
naturally lead to ternary generalizations of the duality concept and the quantum double which are key
ingredients in the theory of quantum groups DRINFELD [1987], KASSEL [1995], MAJID [1995].
14. CONCLUSIONS
In this paper we presented a review of polyadic systems and their representations, ternary algebras
and Hopf algebras. We have classified general polyadic systems and considered their homomorphisms
and their multiplace generalizations, paying attention to their associativity. Then, we defined multi-
place representations and multiactions and have given examples of matrix representations for some
ternary groups. We defined and investigated ternary algebras and Hopf algebras, and have given some
examples. We then considered some ternary generalizations of quantum groups and the Yang-Baxter
equation.
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