Steven Duplij, "Polyadic analogs of direct product"

arXiv:2201.08479v1
[math.RA]
20
Jan
2022
POLYADIC ANALOGS OF DIRECT PRODUCT
STEVEN DUPLIJ
Center for Information Technology (WWU IT), Universität Münster, Röntgenstrasse 7-13
D-48149 Münster, Deutschland
ABSTRACT. We propose a generalization of the external direct product concept to polyadic algebraic struc-
tures which introduces novel properties in two ways: the arity of the product can differ from that of the
constituents, and the elements from different multipliers can be “entangled” such that the product is no
longer componentwise. The main property which we want to preserve is associativity, which is gained by
using the associativity quiver technique provided earlier. For polyadic semigroups and groups we introduce
two external products: 1) the iterated direct product which is componentwise, but can have arity different
from the multipliers; 2) the hetero product (power) which is noncomponentwise and constructed by analogy
with the heteromorphism concept introduced earlier. It is shown in which cases the product of polyadic
groups can itself be a polyadic group. In the same way the external product of polyadic rings and fields is
generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as
opposed to the binary fields), when all multipliers are zeroless fields. Many illustrative concrete examples
are presented.
CONTENTS
1. INTRODUCTION 2
2. PRELIMINARIES 2
3. POLYADIC PRODUCTS OF SEMIGROUPS AND GROUPS 5
3.1. Full polyadic external product 5
3.2. Mixed arity iterated product 6
3.3. Polyadic hetero product 7
4. POLYADIC PRODUCTS OF RINGS AND FIELDS 11
4.1. External direct product of binary rings 11
4.2. Polyadic rings 12
4.3. Full polyadic external direct product of pm, nq-rings 13
4.4. Mixed arity iterated product of pm, nq-rings 14
4.5. Polyadic hetero product of pm, nq-fields 16
REFERENCES 21
E-mail address: douplii@uni-muenster.de; sduplij@gmail.com; https://ivv5hpp.uni-muenster.de/u/douplii.
Date: of completion January 19, 2022. Total: 25 references.
2010 Mathematics Subject Classification. 16T25, 17A42, 20B30, 20F36, 20M17, 20N15.
Key words and phrases. direct product, direct power, polyadic semigroup, arity, polyadic ring, polyadic field.

1. INTRODUCTION
The concept of direct product plays a crucial role for algebraic structures in the study of their internal
constitution and their representation in terms of better known/simpler structures (see, e.g. LANG [1965],
LAMBEK [1966]). The general method of the external direct product construction is to take the Carte-
sian product of the underlying sets and endow it with the operations from the algebraic structures under
consideration. Usually this is an identical repetition of the initial multipliers’ operations componentwise
HUNGERFORD [1974]. In the case of polyadic algebraic structures their arity comes into the game, such
that endowing the product with operations becomes nontrivial in two aspects: the arities of all structures
can be different (but “quantized” and not unique) and the elements from different multipliers can be “en-
tangled” making the product not componentwise. The direct (componentwise) product of n-ary groups
was considered in MICHALSKI [1984a], SHCHUCHKIN [2014]. We propose two corresponding polyadic
analogs (changing arity and “entangling”) of the external direct product which preserve associativity, and
therefore allow us to work out polyadic semigroups, groups, rings and fields.
The direct product is important, especially because it plays the role of a product in a corresponding
category (see, e.g. BORCEUX [1994], MAC LANE [1971]). For instance, the class of all polyadic groups
for objects and polyadic group homomorphisms for morphisms form a category which is well-defined,
because it has the polyadic direct product MICHALSKI [1984b], IANCU [1991] as a product.
We then consider polyadic rings and fields in the same way. Since there exist zeroless polyadic fields
DUPLIJ [2017], the well-known statement (see, e.g. LAMBEK [1966]) of the absence of binary fields
that are a direct product of fields does not generalize. We construct polyadic fields which are products
of zeroless fields, which can lead to a new category of polyadic fields. The proposed constructions are
accompanied by concrete illustrative examples.
2. PRELIMINARIES
We introduce here briefly the usual notation, for details see DUPLIJ [2018]. For a non-empty (un-
derlying) set G the n-tuple (or polyad POST [1940]) of elements is denoted by pg1, . . . , gnq, gi P G,
i “ 1, . . . , n, and the Cartesian product is denoted by Gˆn
”
n
hkkkkkkikkkkkkj
G ˆ . . . ˆ G and consists of all such n-
tuples. For all elements equal to g P G, we denote n-tuple (polyad) by a power pgn
q. To avoid unneeded
indices we denote with one bold letter pgq a polyad for which the number of elements in the n-tuple is
clear from the context, and sometimes we will write
`
gpnq
˘
. On the Cartesian product Gˆn
we define a
polyadic (or n-ary) operation µpnq
: Gˆn
Ñ G such that µpnq
rgs ÞÑ h, where h P G. The operations with
n “ 1, 2, 3 are called unary, binary and ternary.
Recall the definitions of some algebraic structures and their special elements (in the notation of DUPLIJ
[2018]). A (one-set) polyadic algebraic structure G is a set G closed with respect to polyadic operations.
In the case of one n-ary operation µpnq
: Gˆn
Ñ G, it is called polyadic multiplication (or n-ary multi-
plication). A one-set n-ary algebraic structure Mpnq
“
@
G | µpnq
D
or polyadic magma (n-ary magma)
– 2 –

PRELIMINARIES
is a set G closed with respect to one n-ary operation µpnq
and without any other additional structure. In
the binary case Mp2q
was also called a groupoid by Hausmann and Ore HAUSMANN AND ORE [1937]
(and CLIFFORD AND PRESTON [1961]). Since the term “groupoid” was widely used in category theory
for a different construction, the so-called Brandt groupoid BRANDT [1927], BRUCK [1966], Bourbaki
BOURBAKI [1998] later introduced the term “magma”.
Denote the number of iterating multiplications by ℓµ, and call the resulting composition an iterated
product
`
µpnq
˘˝ℓµ
, such that
µ1pn1q
“
`
µpnq
˘˝ℓµ def
“
ℓµ
hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj
µpnq
˝
´
µpnq
˝ . . .
´
µpnq
ˆ idˆpn´1q
¯
. . . ˆ idˆpn´1q
¯
, (2.1)
where the arities are connected by
n1
“ niter “ ℓµ pn ´ 1q ` 1, (2.2)
which gives the length of a iterated polyad pgq in our notation
`
µpnq
˘˝ℓµ
rgs.
A polyadic zero of a polyadic algebraic structure Gpnq
@
G | µpnq
D
is a distinguished element z P G (and
the corresponding 0-ary operation µ
p0q
z ) such that for any pn ´ 1q-tuple (polyad) gpn´1q
P Gˆpn´1q
we have
µpnq
“
gpn´1q
, z
‰
“ z, (2.3)
where z can be on any place in the l.h.s. of (2.3). If its place is not fixed it can be a single zero. 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 iteration (2.1).
A (positive) polyadic power of an element is
gxℓµy
“
`
µpnq
˘˝ℓµ
“
gℓµpn´1q`1
‰
. (2.4)
We define associativity as the invariance of the composition of two n-ary multiplications. An element of
a polyadic algebraic structure g is called ℓµ-nilpotent (or simply nilpotent for ℓµ “ 1), if there exist ℓµ
such that
gxℓµy
“ z. (2.5)
A polyadic (n-ary) identity (or neutral element) of a polyadic algebraic structure is a distinguished element
e (and the corresponding 0-ary operation µ
p0q
e ) such that for any element g P G we have
µpnq
“
g, en´1
‰
“ g, (2.6)
where g can be on any place in the l.h.s. of (2.6).
In polyadic algebraic structures, there exist neutral polyads n P Gˆpn´1q
satisfying
µpnq
rg, ns “ g, (2.7)
where g can be on any of n places in the l.h.s. of (2.7). Obviously, the sequence of polyadic identities
en´1
is a neutral polyad (2.6).
A one-set polyadic algebraic structure
@
G | µpnq
D
is called totally associative, if
`
µpnq
˘˝2
rg, h, us “ µpnq
“
g, µpnq
rhs , u
‰
“ invariant, (2.8)
with respect to placement of the internal multiplication µpnq
rhs in r.h.s. on any of n places, with a fixed
order of elements in the any fixed polyad of p2n ´ 1q elements tp2n´1q
“ pg, h, uq P Gˆp2n´1q
.
– 3 –

PRELIMINARIES
A polyadic semigroup Spnq
is a one-set S one-operation µpnq
algebraic structure in which the n-
ary multiplication is associative, Spnq
“
@
S | µpnq
| associativity (2.8)
D
. A polyadic algebraic structure
Gpnq
“
@
G | µpnq
D
is σ-commutative, if µpnq
“ µpnq
˝ σ, or
µpnq
rgs “ µpnq
rσ ˝ gs , g P Gˆn
, (2.9)
where σ ˝ g “
`
gσp1q, . . . , gσpnq
˘
is a permutated polyad and σ is a fixed element of Sn, the permutation
group on n elements. If (2.9) holds for all σ P Sn, then a polyadic algebraic structure is commutative. A
special type of the σ-commutativity
µpnq
“
g, tpn´2q
, h
‰
“ µpnq
“
h, tpn´2q
, g
‰
, (2.10)
where tpn´2q
P Gˆpn´2q
is any fixed pn ´ 2q-polyad, is called semicommutativity. If an n-ary semigroup
Spnq
is iterated from a commutative binary semigroup with identity, then Spnq
is semicommutative. A
polyadic algebraic structure 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
µpnq
ru, h, ts “ g (2.11)
where h can be on any place, and u, t are polyads of the needed length.
A polyadic algebraic structure which is uniquely i-solvable for all places i “ 1, . . . , n is called a n-ary
(or polyadic) quasigroup Qpnq
“
@
Q | µpnq
| solvability
D
. An associative polyadic quasigroup is called a
n-ary (or polyadic) group. In an n-ary group Gpnq
“
@
G | µpnq
D
the only solution of (2.11) is called a
querelement of g and denoted by ḡ DÖRNTE [1929], such that
µpnq
rh, ḡs “ g, g, ḡ P G, (2.12)
where ḡ can be on any place. Any idempotent g coincides with its querelement ḡ “ g. The unique
solvability relation (2.12) in a n-ary group can be treated as a definition of the unary (multiplicative)
queroperation
µ̄p1q
rgs “ ḡ. (2.13)
We observe from (2.12) and (2.7) that the polyad
ng “
`
gn´2
ḡ
˘
(2.14)
is neutral for any element of a polyadic group, where ḡ can be on any place. If this i-th place is important,
then we write ng;i. In a polyadic group the Dörnte relations DÖRNTE [1929]
µpnq
rg, nh;is “ µpnq
rnh;j, gs “ g (2.15)
hold true for any allowable i, j. In the case of a binary group the relations (2.15) become g ¨ h ¨ h´1
“
h ¨ h´1
¨ g “ g.
Using the queroperation (2.13) one can give a diagrammatic definition of a polyadic group
GLEICHGEWICHT AND GŁAZEK [1967]: an n-ary group is a one-set algebraic structure (universal al-
gebra)
Gpnq
“
@
G | µpnq
, µ̄p1q
| associativity (2.8), Dörnte relations (2.15)
D
, (2.16)
where µpnq
is a n-ary associative multiplication and µ̄p1q
is the queroperation (2.13).
– 4 –

Full polyadic external product POLYADIC PRODUCTS OF SEMIGROUPS AND GROUPS
3. POLYADIC PRODUCTS OF SEMIGROUPS AND GROUPS
We start from the standard external direct product construction for semigroups. Then we show that
consistent “polyadization” of the semigroup direct product, which preserves associativity, can lead to
additional properties:
1) The arities of the polyadic direct product and power can differ from that of the initial semigroups.
(3.1)
2) The components of the polyadic power can contain elements from different multipliers. (3.2)
We use here a vector-like notation for clarity and convenience in passing to higher arity generalizations.
Begin from the direct product of two (binary) semigroups G1,2 ” Gp2q
1,2 “
A
G1,2 | µ
p2q
1,2 ” p¨1,2q | assoc
E
,
where G1,2 are underlying sets, while µ
p2q
1,2 are multiplications in G1,2. On the Cartesian product of the
underlying sets G1
“ G1 ˆ G2 we define a direct product G1 ˆ G2 “ G1
“
@
G1
| µ1p2q
” p‚1
q
D
of the
semigroups G1,2 by the componentwise multiplication of the doubles G “
ˆ
g1
g2
˙
P G1 ˆ G2 (being the
Kronecker product of doubles in our notation) , as
Gp1q
‚1
Gp2q
“
ˆ
g1
g2
˙p1q
‚1
ˆ
g1
g2
˙p2q
“
˜
g
p1q
1 ¨1 g
p2q
1
g
p1q
2 ¨2 g
p2q
2
¸
, (3.3)
and in the “polyadic” notation
µ1p2q
“
Gp1q
, Gp2q
‰
“
¨
˝
µ
p2q
1
”
g
p1q
1 , g
p2q
1
ı
µ
p2q
2
”
g
p1q
2 , g
p2q
2
ı
˛
‚. (3.4)
Obviously, the associativity of µ1p2q
follows immediately from that of µ
p2q
1,2, because of the component-
wise multiplication in (3.4). If G1,2 are groups with the identities e1,2 P G1,2, then the identity of the direct
product is the double E “
ˆ
e1
e2
˙
, such that µ1p2q
rE, Gs “ µ1p2q
rG, Es “ G P G.
3.1. Full polyadic external product. The “polyadization” of (3.4) is straightforward
Definition 3.1. An n1
-ary full direct product semigroup G1pn1q
“ Gpnq
1 ˆGpnq
2 consists of (two or k) n-ary
semigroups (of the same arity n1
“ n)
µ1pnq
“
Gp1q
, Gp2q
, . . . , Gpnq
‰
“
¨
˝
µ
pnq
1
”
g
p1q
1 , g
p2q
1 , . . . , g
pnq
1
ı
µ
pnq
2
”
g
p1q
2 , g
p2q
2 , . . . , g
pnq
2
ı
˛
‚, (3.5)
where the (total) polyadic associativity (2.8) of µ1pn1q
is governed by those of the constituent semigroups
Gpnq
1 and Gpnq
2 (or Gpnq
1 . . .Gpnq
k ) and the componentwise construction (3.5).
If Gpnq
1,2 “
A
G1,2 | µ
pnq
1,2 , µ̄
p1q
1,2
E
are n-ary groups (where µ̄
p1q
1,2 are the unary multiplicative
queroperations (2.13)), then the queroperation µ̄1p1q
of the full direct product group G1pn1q
“
@
G1
” G1 ˆ G2 | µ1pn1q
, µ̄1p1q
D
(n1
“ n) is defined componentwise as follows
Ḡ ” µ̄1p1q
rGs “
˜
µ̄
p1q
1 rg1s
µ̄
p1q
2 rg2s
¸
, or Ḡ “
ˆ
ḡ1
ḡ2
˙
, (3.6)
– 5 –

POLYADIC PRODUCTS OF SEMIGROUPS AND GROUPS Mixed arity iterated product
which satisfies µ1pnq
“
G, G, . . . , Ḡ
‰
“ G with Ḡ on any place (cf. (2.12)).
Definition 3.2. A full polyadic direct product G1pnq
“ Gpnq
1 ˆ Gpnq
2 is called derived, if its constituents
Gpnq
1 and Gpnq
2 are derived, such that the operations µ
pnq
1,2 are compositions of the binary operations µ
p2q
1,2,
correspondingly.
In the derived case all the operations in (3.5) have the form (see (2.1)–(2.2))
µ
pnq
1,2 “
´
µ
p2q
1,2
¯˝pn´1q
, µpnq
“
`
µp2q
˘˝pn´1q
. (3.7)
The operations of the derived polyadic semigroup can be written as (cf. the binary direct product
(3.3)–(3.4))
µ1pnq
“
Gp1q
, Gp2q
, . . . , Gpnq
‰
“ Gp1q
‚1
Gp2q
‚1
. . . ‚1
Gpnq
“
˜
g
p1q
1 ¨1 g
p2q
1 ¨1 . . . ¨1 g
pnq
1
g
p1q
2 ¨2 g
p2q
2 ¨2 . . . ¨2 g
pnq
2
¸
. (3.8)
We will be more interested in nonderived polyadic analogs of the direct product.
Example 3.3. Let us have two ternary groups: the unitless nonderived group Gp3q
1 “
A
iR | µ
p3q
1
E
, where
i2
“ ´1, µ
p3q
1
”
g
p1q
1 , g
p2q
1 , g
p3q
1
ı
“ g
p1q
1 g
p2q
1 g
p3q
1 is a triple product in C, the querelement is µ̄
p1q
1 rg1s “ 1{g1,
and Gp3q
2 “
A
R | µ
p3q
2
E
with µ
p3q
2
”
g
p1q
2 , g
p2q
2 , g
p3q
2
ı
“ g
p1q
2
´
g
p2q
2
¯´1
g
p3q
2 , the querelement µ̄
p1q
2 rg2s “ g2.
Then the ternary nonderived full direct product group becomes G1p3q
“
@
iR ˆ R | µ1p3q
, µ̄1p1q
D
, where
µ1p3q
“
Gp1q
, Gp2q
, Gp3q
‰
“
˜
g
p1q
1 g
p2q
1 g
p3q
1
g
p1q
2
´
g
p2q
2
¯´1
g
p3q
2
¸
, Ḡ ” µ̄1p1q
rGs “
ˆ
1{g1
g2
˙
, (3.9)
which contains no identity, because Gp3q
1 is unitless and nonderived.
3.2. Mixed arity iterated product. In the polyadic case, the following question arises, which cannot
even be stated in the binary case: is it possible to build a version of the associative direct product such
that it can be nonderived and have different arity than the constituent semigroup arities? The answer is yes,
which leads to two arity changing constructions: componentwise and noncomponentwise.
1) Iterated direct product (f). In each of the constituent polyadic semigroups we use the iterating
(2.1) componentwise, but with different numbers of compositions, because the same number of
compositions evidently leads to the iterated polyadic direct product. In this case the arity of the
direct product is greater than or equal to the arities of the constituents n1
ě n1, n2.
2) Hetero product (b). The polyadic product of k copies of the same n-ary semigroup is constructed
using the associativity quiver technique, which mixes (”entangles”) elements from different multi-
pliers, it is noncomponentwise (by analogy with heteromorphisms in DUPLIJ [2018]), and so it can
be called a hetero product or hetero power (for coinciding multipliers, i.e. constituent polyadic
semigroups or groups). This gives the arity of the hetero product which is less than or equal to the
arities of the equal multipliers n1
ď n.
In the first componentwise case 1), the constituent multiplications (3.5) are composed from the lower
arity ones in the componentwise way, but the initial arities of up and down components can be different
(as opposed to the binary derived case (3.7))
µ
pnq
1 “
´
µ
pn1q
1
¯˝ℓµ1
, µ
pnq
2 “
´
µ
pn2q
2
¯˝ℓµ2
, 3 ď n1,2 ď n ´ 1, (3.10)
– 6 –

Polyadic hetero product POLYADIC PRODUCTS OF SEMIGROUPS AND GROUPS
where we exclude the limits: the derived case n1,2 “ 2 (3.7) and the undecomposed case n1,2 “ n (3.5).
Since the total size of the up and down polyads is the same and coincides with the arity of the double G
multiplication n1
, using (2.2) we obtain the arity compatibility relations
n1
“ ℓµ1 pn1 ´ 1q ` 1 “ ℓµ2 pn2 ´ 1q ` 1. (3.11)
Definition 3.4. A mixed arity polyadic iterated direct product semigroup G1pn1q
“ Gpn1q
1 fGpn2q
2 consists
of (two) polyadic semigroups Gpn1q
1 and Gpn2q
2 of the different arity shapes n1 and n2
µ1pn1q
”
Gp1q
, Gp2q
, . . . , Gpn1q
ı
“
¨
˚
˝
´
µ
pn1q
1
¯˝ℓµ1
”
g
p1q
1 , g
p2q
1 , . . . , g
pnq
1
ı
´
µ
pn2q
2
¯˝ℓµ2
”
g
p1q
2 , g
p2q
2 , . . . , g
pnq
2
ı
˛
‹
‚, (3.12)
and the arity compatibility relations (3.11) hold.
Observe that it is not the case that any two polyadic semigroups can be composed in the mixed arity
polyadic direct product.
Assertion 3.5. If the arity shapes of two polyadic semigroups Gpn1q
1 and Gpn2q
2 satisfy the compatibility
condition
a pn1 ´ 1q “ b pn2 ´ 1q “ c, a, b, c P N, (3.13)
then they can form a mixed arity direct product G1pn1q
“ Gpn1q
1 f Gpn2q
2 , where n1
“ c ` 1 (3.11).
Example 3.6. In the case of a 4-ary and 5-ary semigroups Gp4q
1 and Gp5q
2 the direct product arity of G1pn1q
is “quantized” n1
“ 3ℓµ1 ` 1 “ 4ℓµ2 ` 1, such that
n1
“ 12k ` 1 “ 13, 25, 37, . . ., (3.14)
ℓµ1 “ 4k “ 4, 8, 12, . . ., (3.15)
ℓµ2 “ 3k “ 3, 6, 9, . . ., k P N, (3.16)
and only the first mixed arity 13-ary direct product semigroup G1p13q
is nonderived. If Gp4q
1 and Gp5q
2 are
polyadic groups with the queroperations µ̄
p1q
1 and µ̄
p1q
2 correspondingly, then the iterated direct G1pn1q
is a
polyadic group with the queroperation µ̄1p1q
given in (3.6).
In the same way one can consider the iterated direct product of any number of polyadic semigroups.
3.3. Polyadic hetero product. In the second noncomponentwise case 2) we allow multiplying elements
from different components, and therefore we should consider the Cartesian k-power of sets G1
“ Gˆk
and endow the corresponding k-tuple with a polyadic operation in such a way that associativity of Gpnq
will govern the associativity of the product G1pnq
. In other words we construct a k-power of the polyadic
semigroup Gpnq
such that the result G1pn1q
is an n1
-ary semigroup.
The general structure of the hetero product formally coincides “reversely” with the main heteromor-
phism equation DUPLIJ [2018]. The additional parameter which determines the arity n1
of the hetero
power of the initial n-ary semigroup is the number of intact elements ℓid. Thus, we arrive at
Definition 3.7. The hetero (“entangled”) k-power of the n-ary semigroup Gpnq
“
@
G | µpnq
D
is the n1
-ary
semigroup defined on the k-th Cartesian power G1
“ Gˆk
, such that G1pn1q
“
@
G1
| µ1pn1q
D
,
G1pn1q
“
`
Gpnq
˘bk
”
k
hkkkkkkkkkikkkkkkkkkj
Gpnq
b . . . b Gpnq
, (3.17)
– 7 –

POLYADIC PRODUCTS OF SEMIGROUPS AND GROUPS Polyadic hetero product
TABLE 1. Hetero power “quantization”.
k ℓµ ℓid n{n1
2 1 1
n “ 3, 5, 7, . . .
n1
“ 2, 3, 4, . . .
3 1 2
n “ 4, 7, 10, . . .
n1
“ 2, 3, 4, . . .
3 2 1
n “ 4, 7, 10, . . .
n1
“ 3, 5, 7, . . .
4 1 3
n “ 5, 9, 13, . . .
n1
“ 2, 3, 4, . . .
4 2 2
n “ 3, 5, 7, . . .
n1
“ 2, 3, 4, . . .
4 3 1
n “ 5, 9, 13, . . .
n1
“ 4, 7, 10, . . .
and the n1
-ary multiplication of k-tuples GT
“ pg1, g2, . . . , gkq P Gˆk
is given (informally) by
µ1pn1q
»
–
¨
˝
g1
.
.
.
gk
˛
‚, . . . ,
¨
˝
gkpn1´1q
.
.
.
gkn1
˛
‚
ﬁ
ﬂ “
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
µpnq
rg1, . . . , gns ,
.
.
.
µpnq
“
gnpℓµ´1q, . . . , gnℓµ
‰
,
/
.
/
-
ℓµ
gnℓµ`1,
.
.
.
gnℓµ`ℓid
,
.
-
ℓid
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
, gi P G, (3.18)
where ℓid is the number of intact elements in the r.h.s., and ℓµ “ k ´ ℓid is the number of multiplications
in the resulting k-tuple of the direct product. The hetero power parameters are connected by the arity
changing formula DUPLIJ [2018]
n1
“ n ´
n ´ 1
k
ℓid, (3.19)
with the integer
n ´ 1
k
ℓid ě 1.
The concrete placement of elements and multiplications in (3.18) to obtain the associative µ1pn1q
is
governed by the associativity quiver technique DUPLIJ [2018].
There exist important general numerical relations between the parameters of the twisted direct power
n1
, n, k, ℓid, which follow from (3.18)–(3.19). First, there are non-strict inequalities for them
0 ď ℓid ď k ´ 1, (3.20)
ℓµ ď k ď pn ´ 1q ℓµ, (3.21)
2 ď n1
ď n. (3.22)
Second, the initial and final arities n and n1
are not arbitrary, but “quantized” such that the fraction in
(3.19) has to be an integer (see TABLE 1).
Assertion 3.8. The hetero power is not unique in both directions, if we do not fix the initial n and final n1
arities of Gpnq
and G1pn1q
.
Proof. This follows from (3.18) and the hetero power “quantization” TABLE 1.
– 8 –

Polyadic hetero product POLYADIC PRODUCTS OF SEMIGROUPS AND GROUPS
The classification of the hetero powers consists of two limiting cases.
1) Intactless power: there are no intact elements ℓid “ 0. The arity of the hetero power reaches its
maximum and coincides with the arity of the initial semigroup n1
“ n (see Example 3.12).
2) Binary power: the final semigroup is of lowest arity, i.e. binary n1
“ 2. The number of intact
elements is (see Example 3.11)
ℓid “ k
n ´ 2
n ´ 1
. (3.23)
Example 3.9. Consider the cubic power of a 4-ary semigroup G1p3q
“
`
Gp4q
˘b3
with the identity e, then
the ternary identity triple in G1p3q
is ET
“ pe, e, eq, and therefore this cubic power is a ternary semigroup
with identity.
Proposition 3.10. If the initial n-ary semigroup Gpnq
contains an identity, then the hetero power G1pn1q
“
`
Gpnq
˘bk
can contain an identity in the intactless case and the Post-like quiver DUPLIJ [2018]. For the
binary power k “ 2 only the one-sided identity is possible.
Let us consider some concrete examples.
Example 3.11. Let Gp3q
“
@
G | µp3q
D
be a ternary semigroup, then we can construct its power k “ 2
(square) of the doubles G “
ˆ
g1
g2
˙
P G ˆ G “ G1
in two ways to obtain the associative hetero power
µ1p2q
“
Gp1q
, Gp2q
‰
“
$
’
’
’
’

’
’
’
’
%
˜
µp3q
”
g
p1q
1 , g
p1q
2 , g
p2q
1
ı
g
p2q
2
¸
,
˜
µp3q
”
g
p1q
1 , g
p2q
2 , g
p2q
1
ı
g
p1q
2
¸
,
g
pjq
i P G. (3.24)
This means that the Cartesian square can be endowed with the associative multiplication µ1p2q
, and there-
fore G1p2q
“
@
G1
| µ1p2q
D
is a binary semigroup being the hetero product G1p2q
“ Gp3q
b Gp3q
. If Gp3q
has a ternary identity e P G, then G1p2q
has only the left (right) identity E “
ˆ
e
e
˙
P G1
, since
µ1p2q
rE, Gs “ G (µ1p2q
rG, Es “ G), but not the right (left) identity. Thus, G1p2q
can be a semigroup
only, even if Gp3q
is a ternary group.
Example 3.12. Take Gp3q
“
@
G | µp3q
D
a ternary semigroup, then the multiplication on the double G “
ˆ
g1
g2
˙
P G ˆ G “ G1
is ternary and noncomponentwise
µ1p3q
“
Gp1q
, Gp2q
, Gp3q
‰
“
¨
˝
µp3q
”
g
p1q
1 , g
p2q
2 , g
p3q
1
ı
µp3q
”
g
p1q
2 , g
p2q
1 , g
p3q
2
ı
˛
‚, g
pjq
i P G, (3.25)
and µ1p3q
is associative (and described by the Post-like associative quiver DUPLIJ [2018]), and therefore the
cubic hetero power is the ternary semigroup G1p3q
“
@
G ˆ G | µ1p3q
D
, such that G1p3q
“ Gp3q
b Gp3q
. In
this case, as opposed to the previous example, the existence of a ternary identity in Gp3q
implies the ternary
identity in the direct cube G1p3q
by E “
ˆ
e
e
˙
. If Gp3q
is a ternary group with the unary queroperation
µ̄p1q
, then the cubic hetero power G1p3q
is also a ternary group of the special class DUDEK [1990]: all
– 9 –

POLYADIC PRODUCTS OF SEMIGROUPS AND GROUPS Polyadic hetero product
querelements coincide (cf. (3.6)), such that ḠT
“ pgquer, gquerq, where µ̄p1q
rgs “ gquer, @g P G. This is
because in (2.12) the querelement can be on any place.
Theorem 3.13. If Gpnq
is an n-ary group, then the hetero k-power G1pn1q
“
`
Gpnq
˘bk
can contain quer-
operations in the intactless case only.
Corollary 3.14. If the power multiplication (3.18) contains no intact elements ℓid “ 0 and does not change
arity n1
“ n, a hetero power can be a polyadic group which has only one querelement.
Next we consider more complicated hetero power (“entangled”) constructions with and without intact
elements, as well as Post-like and non-Post associative quivers DUPLIJ [2018].
“
@
G | µp4q
D
be a 4-ary semigroup, then we can construct its 4-ary associative
cubic hetero power G1p4q
“
@
G1
| µ1p4q
D
using the Post-like and non-Post associative quivers without
intact elements. Taking in (3.18) n1
“ n, k “ 3, ℓid “ 0, we get two possibilities for the multiplication of
the triples GT
“ pg1, g2, g3q P G ˆ G ˆ G “ G1
1) Post-like associative quiver. The multiplication of the hetero cubic power case takes the form
µ1p4q
“
Gp1q
, Gp2q
, Gp3q
, Gp4q
‰
“
¨
˚
˚
˚
˝
µp4q
”
g
p1q
1 , g
p2q
2 , g
p3q
3 , g
p4q
1
ı
µp4q
”
g
p1q
2 , g
p2q
3 , g
p3q
1 , g
p4q
2
ı
µp4q
”
g
p1q
3 , g
p2q
1 , g
p3q
2 , g
p4q
3
ı
˛
‹
‹
‹
‚
, g
pjq
i P G, (3.26)
and it can be shown that µ1p4q
is totally associative, therefore G1p4q
“
@
G1
| µ1p4q
D
is a 4-ary semi-
group.
2) Non-Post associative quiver. The multiplication of the hetero cubic power differs from (3.26)
µ1p4q
“
Gp1q
, Gp2q
, Gp3q
, Gp4q
‰
“
¨
˚
˚
˚
˝
µp4q
”
g
p1q
1 , g
p2q
3 , g
p3q
2 , g
p4q
1
ı
µp4q
”
g
p1q
2 , g
p2q
1 , g
p3q
3 , g
p4q
2
ı
µp4q
”
g
p1q
3 , g
p2q
2 , g
p3q
1 , g
p4q
3
ı
˛
‹
‹
‹
‚
, g
pjq
i P G, (3.27)
and it can be shown that µ1p4q
is totally associative, therefore G1p4q
“
@
G1
| µ1p4q
D
is a 4-ary semi-
group.
The following is valid for both the above cases. If Gp4q
has the 4-ary identity satisfying
µp4q
re, e, e, gs “ µp4q
re, e, g, es “ µp4q
re, g, e, es “ µp4q
rg, e, e, es “ g, @g P G, (3.28)
then the hetero power G1p4q
has the 4-ary identity
E “
¨
˝
e
e
e
˛
‚, e P G. (3.29)
In the case where Gp3q
is a ternary group with the unary queroperation µ̄p1q
, then the cubic hetero power
G1p4q
is also a ternary group with one querelement (cf. Example 3.12)
Ḡ “
¨
˝
g1
g2
g3
˛
‚“
¨
˝
gquer
gquer
gquer
˛
‚, gquer P G, gi P G, (3.30)
where gquer “ µ̄p1q
rgs, @g P G.
– 10 –

External direct product of binary rings POLYADIC PRODUCTS OF SEMIGROUPS AND GROUPS
A more nontrivial example is a cubic hetero power which has different arity to the initial semigroup.
“
@
G | µp4q
D
be a 4-ary semigroup, then we can construct its ternary associative
cubic hetero power G1p3q
“
@
G1
| µ1p3q
D
using the associative quivers with one intact element and two
multiplications DUPLIJ [2018]. Taking in (3.18) the parameters n1
“ 3, n “ 4, k “ 3, ℓid “ 1 (see
third line of TABLE 1), we get for the ternary multiplication µ1p3q
for the triples GT
“ pg1, g2, g3q P
G ˆ G ˆ G “ G1
of the hetero cubic power case the form
µ1p3q
“
Gp1q
, Gp2q
, Gp3q
‰
“
¨
˚
˚
˝
µp4q
”
g
p1q
1 , g
p2q
2 , g
p3q
3 , g
p3q
1
ı
µp4q
”
g
p1q
2 , g
p2q
3 , g
p2q
1 , g
p3q
2
ı
g
p1q
3
˛
‹
‹
‚, g
pjq
i P G, (3.31)
which is totally associative, and therefore the hetero cubic power of 4-ary semigroup Gp4q
“
@
G | µp4q
D
is
a ternary semigroup G1p3q
“
@
G1
| µ1p3q
D
, such that G1p3q
“
`
Gp4q
˘b 3
. If the initial 4-ary semigroup Gp4q
has the identity satisfying (3.28), then the ternary hetero power G1p3q
has only the right ternary identity
(3.29) satisfying one relation
µ1p3q
rG, E, Es “ G, @G P Gˆ3
, (3.32)
and therefore G1p3q
is a ternary semigroup with a right identity. If Gp4q
is a 4-ary group with the quer-
operation µ̄p1q
, then the hetero power G1p3q
can only be a ternary semigroup , because in
@
G1
| µ1p3q
D
we
cannot define the standard queroperation POST [1940].
4. POLYADIC PRODUCTS OF RINGS AND FIELDS
Now we show that the thorough “polyadization” of operations can lead to some unexpected new prop-
erties of ring and field external direct products. Recall that in the binary case the external direct product
of fields does not exist at all (see, e.g., LAMBEK [1966]). The main new peculiarities of the polyadic case
are:
1) The arity shape of the external product ring and its constituent rings can be different.
2) The external product of polyadic fields can be a polyadic field.
4.1. External direct product of binary rings. First, we recall the ordinary (binary) direct product of
rings in notation which would be convenient to generalize to higher arity structures DUPLIJ [2017]. Let
us have two binary rings R1,2 ” Rp2,2q
1,2 “
A
R1,2 | ν
p2q
1,2 ” p`1,2q , µ
p2q
1,2 ” p¨1,2q
E
, where R1,2 are un-
derlying sets, while ν
p2q
1,2 and µ
p2q
1,2 are additions and multiplications (satisfying distributivity) in R1,2,
correspondingly. On the Cartesian product of the underlying sets R1
“ R1 ˆ R2 one defines the external
direct product ring R1 ˆR2 “ R1
“
@
R1
| ν1p2q
” p`1
q , µ1p2q
” p‚1
q
D
by the componentwise operations
(addition and multiplication) on the doubles X “
ˆ
x1
x2
˙
P R1 ˆ R2 as follows
Xp1q
`1
Xp2q
“
ˆ
x1
x2
˙p1q
`1
ˆ
x1
x2
˙p2q
”
˜
x
p1q
1
x
p1q
2
¸
`1
˜
x
p2q
1
x
p2q
2
¸
“
˜
x
p1q
1 `1 x
p2q
1
x
p1q
2 `2 x
p2q
2
¸
, (4.1)
Xp1q
‚1
Xp2q
“
ˆ
x1
x2
˙p1q
‚1
ˆ
x1
x2
˙p2q
“
˜
x
p1q
1 ¨1 x
p2q
1
x
p1q
2 ¨2 x
p2q
2
¸
, (4.2)
– 11 –

POLYADIC PRODUCTS OF RINGS AND FIELDS Polyadic rings
or in the polyadic notation (with manifest operations)
ν1p2q
“
Xp1q
, Xp2q
‰
“
¨
˝
ν
p2q
1
”
x
p1q
1 , x
p2q
1
ı
ν
p2q
2
”
x
p1q
2 , x
p2q
2
ı
˛
‚, (4.3)
µ1p2q
“
Xp1q
, Xp2q
‰
“
¨
˝
µ
p2q
1
”
x
p1q
1 , x
p2q
1
ı
µ
p2q
2
”
x
p1q
2 , x
p2q
2
ı
˛
‚. (4.4)
The associativity and distributivity of the binary direct product operations ν1p2q
and µ1p2q
are obviously
governed by those of the constituent binary rings R1 and R2, because of the componentwise construc-
tion on the r.h.s. of (4.3)–(4.4). In the polyadic case, the construction of the direct product is not so
straightforward and can have additional unusual peculiarities.
4.2. Polyadic rings. Here we recall definitions of polyadic rings ČUPONA [1965], CROMBEZ [1972],
LEESON AND BUTSON [1980] in our notation DUPLIJ [2017, 2018]. Consider a polyadic structure
@
R | µpnq
, νpmq
D
with two operations on the same set R: the m-ary addition νpmq
: Rˆm
Ñ R and
the n-ary multiplication µpnq
: Rˆn
Ñ R. The “interaction” between operations can be defined using the
polyadic analog of distributivity.
Definition 4.1. The polyadic distributivity for µpnq
and νpmq
consists of n relations
µpnq
“
νpmq
rx1, . . . xms , y2, y3, . . . yn
‰
“ νpmq
“
µpnq
rx1, y2, y3, . . . yns , µpnq
rx2, y2, y3, . . . yns , . . . µpnq
rxm, y2, y3, . . . yns
‰
(4.5)
µpnq
“
y1, νpmq
rx1, . . . xms , y3, . . . yn
‰
“ νpmq
“
µpnq
ry1, x1, y3, . . . yns , µpnq
ry1, x2, y3, . . . yns , . . . µpnq
ry1, xm, y3, . . . yns
‰
(4.6)
.
.
.
µpnq
“
y1, y2, . . . yn´1, νpmq
rx1, . . . xms
‰
“ νpmq
“
µpnq
ry1, y2, . . . yn´1, x1s , µpnq
ry1, y2, . . . yn´1, x2s , . . . µpnq
ry1, y2, . . . yn´1, xms
‰
, (4.7)
where xi, yj P R.
The operations µpnq
and νpmq
are totally associative, if (in the invariance definition DUPLIJ [2017,
2018])
νpmq
“
u, νpmq
rvs , w
‰
µpnq
“
x, µpnq
rys , t
‰
where the internal products can be on any place, and y P Rˆn
, v P Rˆm
, and the polyads x, t, u, w are
of the needed lengths. In this way both algebraic structures
@
R | µpnq
| assoc
D
and
@
R | νpmq
| assoc
D
are
polyadic semigroups Spnq
and Spmq
.
Definition 4.2. A polyadic pm, nq-ring Rpm,nq
is a set R with two operations µpnq
: Rˆn
Ñ R and
νpmq
: Rˆm
Ñ R, such that:
1) they are distributive (4.5)-(4.7);
2)
@
R | µpnq
| assoc
D
is a polyadic semigroup;
3)
@
R | νpmq
| assoc, comm, solv
D
is a commutative polyadic group.
– 12 –

Full polyadic external direct product of pm, nq-rings POLYADIC PRODUCTS OF RINGS AND FIELDS
In case the multiplicative semigroup
@
R | µpnq
| assoc
D
of Rpm,nq
is commutative, µpnq
rxs “
µpnq
rσ ˝ xs, for all σ P Sn, then Rpm,nq
is called a commutative polyadic ring, and if it contains the
identity, then Rpm,nq
is a pm, nq-semiring. A polyadic ring Rpm,nq
is called derived, if νpmq
and µpnq
are
repetitions of the binary addition p`q and multiplication p¨q, while xR | p`qy and xR | p¨qy are commuta-
tive (binary) group and semigroup respectively.
4.3. Full polyadic external direct product of pm, nq-rings. Let us consider the following task: for a
given polyadic pm, nq-ring R1pm,nq
“
@
R1
| ν1pmq
, µ1pnq
D
to construct a product of all possible (in arity
shape) constituent rings Rpm1,n1q
1 and Rpm2,n2q
2 . The first-hand “polyadization” of (4.3)–(4.4) leads to
Definition 4.3. A full polyadic direct product ring R1pm,nq
“ Rpm,nq
1 ˆRpm,nq
2 consists of (two) polyadic
rings of the same arity shape, such that
ν1pmq
“
Xp1q
, Xp2q
, . . . , Xpmq
‰
“
¨
˝
ν
pmq
1
”
x
p1q
1 , x
p2q
1 , . . . , x
pmq
1
ı
ν
pmq
2
”
x
p1q
2 , x
p2q
2 , . . . , x
pmq
2
ı
˛
‚, (4.10)
µ1pnq
“
Xp1q
, Xp2q
, . . . , Xpnq
‰
“
¨
˝
µ
pnq
1
”
x
p1q
1 , x
p2q
1 , . . . , x
pnq
1
ı
µ
pnq
2
”
x
p1q
2 , x
p2q
2 , . . . , x
pnq
2
ı
˛
‚, (4.11)
where the polyadic associativity (2.8) and polyadic distributivity (4.5)–(4.7) of the direct product oper-
ations νpmq
and µpnq
follow from those of the constituent rings and the componentwise operations in
(4.10)–(4.11).
Example 4.4. Consider two p2, 3q-rings Rp2,3q
1 “
A
tixu | ν
p2q
1 “ p`q , µ
p3q
1 “ p¨q , x P Z, i2
“ ´1
E
and
Rp2,3q
2 “
Bˆ
0 a
b 0
˙*
| ν
p2q
2 “ p`q , µ
p3q
2 “ p¨q , a, b P Z
F
, where p`q and p¨q are operations in Z, then
their polyadic direct product on the doubles XT
“
ˆ
ix,
ˆ
0 a
b 0
˙˙
P
`
iZ, GLadiag
p2, Zq
˘
is defined by
ν1p2q
“
Xp1q
, Xp2q
‰
“
¨
˝
ixp1q
` ixp2q
ˆ
0 ap1q
` ap2q
bp1q
` bp2q
0
˙
˛
‚, (4.12)
µ1p3q
“
Xp1q
, Xp2q
, Xp3q
‰
“
¨
˝
ixp1q
xp2q
xp3q
ˆ
0 ap1q
bp2q
ap3q
bp1q
ap2q
bp3q
0
˙
˛
‚. (4.13)
The polyadic associativity and distributivity of the direct product operations ν1p2q
and µ1p3q
are evident,
and therefore Rp2,3q
“
Bˆ
ix,
ˆ
0 a
b 0
˙˙*
| ν1p2q
, µ1p3q
F
is a p2, 3q-ring Rp2,3q
“ Rp2,3q
1 ˆ Rp2,3q
2 .
Definition 4.5. A polyadic direct product Rpm,nq
is called derived, if both constituent rings Rpm,nq
1 and
Rpm,nq
2 are derived, such that the operations ν
pmq
1,2 and µ
pnq
1,2 are compositions of the binary operations ν
p2q
1,2
and µ
p2q
1,2, correspondingly.
– 13 –

POLYADIC PRODUCTS OF RINGS AND FIELDS Mixed arity iterated product of pm, nq-rings
So, in the derived case (see (2.1) all the operations in (4.10)–(4.11) have the form (cf. (3.7))
ν
pmq
1,2 “
´
ν
p2q
1,2
¯˝pm´1q
, µ
pnq
1,2 “
´
ν
p2q
1,2
¯˝pn´1q
, (4.14)
νpmq
“
`
νp2q
˘˝pm´1q
, µpnq
“
`
νp2q
˘˝pn´1q
. (4.15)
Thus, the operations of the derived polyadic ring can be written as (cf. the binary direct product (4.1)–
(4.2))
ν1pmq
“
Xp1q
, Xp2q
, . . . , Xpmq
‰
“
˜
x
p1q
1 `1 x
p2q
1 `1 . . . `1 x
pmq
1
x
p1q
2 `2 x
p2q
2 `2 . . . `2 x
pmq
2
¸
, (4.16)
µ1pnq
“
Xp1q
, Xp2q
, . . . , Xpnq
‰
“
˜
x
p1q
1 ¨1 x
p2q
1 ¨1 . . . ¨1 x
pnq
1
x
p1q
2 ¨2 x
p2q
2 ¨2 . . . ¨2 x
pnq
2
¸
, (4.17)
The external direct product p2, 3q-ring Rp2,3q
from Example 4.4 is not derived, because both multipli-
cations µ
p3q
1 and µ
p3q
2 there are nonderived.
4.4. Mixed arity iterated product of pm, nq-rings. Recall, that some polyadic multiplications can be
iterated, i. e. composed (2.1) from those of lower arity (2.2), also larger than 2, and so being nonderived, in
general. The nontrivial “polyadization” of (4.3)–(4.4) can arise, when the composition of the separate (up
and down) components in the r.h.s. of (4.10)–(4.11) will be different, and therefore the external product
operations on the doubles X P R1 ˆ R2 cannot be presented in the iterated form (2.1).
Let now the constituent operations in (4.10)–(4.11) be composed from lower arity corresponding oper-
ations, but in different ways for the up and down components, such that
ν
pmq
1 “
´
ν
pm1q
1
¯˝ℓν1
, ν
pmq
2 “
´
ν
pm2q
2
¯˝ℓν2
, 3 ď m1,2 ď m ´ 1, (4.18)
µ
pnq
1 “
´
µ
pn1q
1
¯˝ℓµ1
, µ
pnq
2 “
´
µ
pn2q
2
¯˝ℓµ2
, 3 ď n1,2 ď n ´ 1, (4.19)
where we exclude the limits: the derived case m1,2 “ n1,2 “ 2 (4.14)–(4.15) and the uncomposed case
m1,2 “ m, n1,2 “ n (4.10)–(4.11). Since the total size of the up and down polyads is the same and
coincides with the arities of the double addition m and multiplication n, using (2.2) we obtain the arity
compatibility relations
m “ ℓν1 pm1 ´ 1q ` 1 “ ℓν2 pm2 ´ 1q ` 1, (4.20)
n “ ℓµ1 pn1 ´ 1q ` 1 “ ℓµ2 pn2 ´ 1q ` 1. (4.21)
Definition 4.6. A mixed arity polyadic direct product ring Rpm,nq
“ Rpm1,n1q
1 f Rpm2,n2q
2 consists of
two polyadic rings of the different arity shape, such that
ν1pmq
“
Xp1q
, Xp2q
, . . . , Xpmq
‰
“
¨
˚
˝
´
ν
pm1q
1
¯˝ℓν1
”
x
p1q
1 , x
p2q
1 , . . . , x
pmq
1
ı
´
ν
pm2q
2
¯˝ℓν2
”
x
p1q
2 , x
p2q
2 , . . . , x
pmq
2
ı
˛
‹
‚, (4.22)
µ1pnq
“
Xp1q
, Xp2q
, . . . , Xpnq
‰
“
¨
˚
˝
´
µ
pn1q
1
¯˝ℓµ1
”
x
p1q
1 , x
p2q
1 , . . . , x
pnq
1
ı
´
µ
pn2q
2
¯˝ℓµ2
”
x
p1q
2 , x
p2q
2 , . . . , x
pnq
2
ı
˛
‹
‚, (4.23)
and the arity compatibility relations (4.20)–(4.21) hold valid.
– 14 –

Mixed arity iterated product of pm, nq-rings POLYADIC PRODUCTS OF RINGS AND FIELDS
Thus, two polyadic rings cannot always be composed in the mixed arity polyadic direct product.
Assertion 4.7. If the arity shapes of two polyadic rings Rpm1,n1q
1 and Rpm2,n2q
2 satisfies the compatibility
conditions
a pm1 ´ 1q “ b pm2 ´ 1q , (4.24)
c pn1 ´ 1q “ d pn2 ´ 1q , a, b, c, d P N, (4.25)
then they can form a mixed arity direct product.
The limiting cases, undecomposed (4.10)–(4.11) and derived (4.16)–(4.17), satisfy the compatibility
conditions (4.24)–(4.25) as well.
Example 4.8. Let us consider two (nonderived) polyadic rings Rp9,3q
1 “
A
t8l ` 7u | ν
p9q
1 , µ
p3q
1 , l P Z
E
and
Rp5,5q
2 “
A
tMu | ν
p5q
2 , µ
p5q
2
E
, where
M “
¨
˚
˚
˝
0 4k1 ` 3 0 0
0 0 4k2 ` 3 0
0 0 0 4k3 ` 3
4k4 ` 3 0 0 0
˛
‹
‹
‚, ki P Z, (4.26)
and ν
p5q
2 and µ
p5q
2 are the ordinary sum and product of 5 matrices. Using (4.20)–(4.21) we obtain m “ 9,
n “ 5, if we choose the smallest “numbers of multiplications” ℓν1 “ 1, ℓν2 “ 2, ℓµ1 “ 2, ℓµ2 “ 1, and
therefore the mixed arity direct product nonderived p9, 5q-ring becomes
Rp9,5q
“
@
tXu | ν1p9q
, µ1p5q
D
, (4.27)
where the doubles are X “
ˆ
8l ` 7
M
˙
and the nonderived direct product operations are
ν1p9q
“
Xp1q
, Xp2q
, . . . , Xp9q
‰
“
¨
˚
˚
˚
˚
˝
8
`
lp1q
` lp2q
` lp3q
` lp4q
` lp5q
` lp6q
` lp7q
` lp8q
` lp9q
` 7
˘
` 7
¨
˚
˚
˝
0 4K1 ` 3 0 0
0 0 4K2 ` 3 0
0 0 0 4K3 ` 3
4K4 ` 3 0 0 0
˛
‹
‹
‚
˛
‹
‹
‹
‹
‚
,
(4.28)
µ1p5q
“
Xp1q
, Xp2q
, Xp3q
, Xp4q
, Xp5q
‰
“
¨
˚
˚
˚
˚
˝
p8lµ ` 7q
¨
˚
˚
˝
0 4Kµ,1 ` 3 0 0
0 0 4Kµ,2 ` 3 0
0 0 0 4Kµ,3 ` 3
4Kµ,4 ` 3 0 0 0
˛
‹
‹
‚
˛
‹
‹
‹
‹
‚
,
(4.29)
where in the first line Ki “ k
p1q
i ` k
p2q
i ` k
p3q
i ` k
p4q
i ` k
p5q
i ` k
p6q
i ` k
p7q
i ` k
p8q
i ` k
p9q
i ` 6 P Z, lµ P Z is a
cumbersome integer function of lpjq
P Z, j “ 1, . . . , 9, and in the second line Kµ,i P Z are cumbersome
integer functions of k
psq
i , i “ 1, . . . , 4, s “ 1, . . . , 5. Therefore the polyadic ring (4.27) is the nonderived
mixed arity polyadic external product Rp9,5q
“ Rp9,3q
1 f Rp5,5q
2 (see Definition 4.6).
Theorem 4.9. The category of polyadic rings PolRing can exist (having the class of all polyadic rings
for objects and ring homomorphisms for morphisms) and can be well-defined, because it has a product
as the polyadic external product of rings.
– 15 –

POLYADIC PRODUCTS OF RINGS AND FIELDS Polyadic hetero product of pm, nq-fields
In the same way one can construct the iterated full and mixed arity products of any number k of polyadic
rings, just by passing from the doubles X to k-tuples XT
k “ px1, . . . , xkq.
4.5. Polyadic hetero product of pm, nq-fields. The most crucial difference between the binary direct
products and the polyadic ones arises for fields, because a direct product two binary fields is not a field
LAMBEK [1966]. The reason lies in the fact that each binary field Fp2,2q
necessarily contains 0 and 1,
by definition. As follows from (4.2), a binary direct product contains nonzero idempotent doubles
ˆ
1
0
˙
and
ˆ
0
1
˙
which are noninvertible, and therefore the external direct product of fields Fp2,2q
1 ˆ Fp2,2q
2
can never be a field. As opposite, polyadic fields (see Definition 4.10) can be zeroless (we denote them
by hat x
F), and the above arguments do not hold valid for them.
Recall definitions of pm, nq-fields (see LEESON AND BUTSON [1980], IANCU AND POP [1997]).
Denote R˚
“ Rz tzu, if the zero z exists (2.3). Observe that (in distinction to binary rings)
@
R˚
| µpnq
| assoc
D
is not a polyadic group, in general. If
@
R˚
| µpnq
D
is the n-ary group, then Rpm,nq
is
called a pm, nq-division ring Dpm,nq
.
Definition 4.10. A (totally) commutative pm, nq-division ring Rpm,nq
is called a pm, nq-field Fpm,nq
.
In n-ary groups there exists an “intermediate” commutativity, so called semicommutativity (2.10).
Definition 4.11. A semicommutative pm, nq-division ring Rpm,nq
is called a semicommutative pm, nq-
field Fpm,nq
.
The definition of a polyadic field can be done in a diagrammatic form, analogous to (2.16). We introduce
the double Dörnte relations: for n-ary multiplication µpnq
(2.15) and for m-ary addition νpmq
, as follows
νpmq
rmy, xs “ x, (4.30)
where the (additive) neutral sequence is my “ pym´2
, ỹq, and ỹ is the additive querelement for y P R (see
(2.14)). As distinct from (2.15) we have only one (additive) Dörnte relation (4.30) and one diagram from
(2.16) only, because of commutativity of νpmq
.
By analogy with the multiplicative queroperation µ̄p1q
(2.13), introduce the additive unary queropera-
tion by
ν̃p1q
pxq “ x̃, @x P R, (4.31)
where x̃ is the additive querelement (2.13). Thus, we have
Definition 4.12 (Diagrammatic definition of pm, nq-field). A (polyadic) pm, nq-field is a one-set algebraic
structure with 4 operations and 3 relations
@
R | νpmq
, ν̃p1q
, µpnq
, µ̄p1q
| associativity, distributivity, double Dörnte relations
D
, (4.32)
where νpmq
and µpnq
are commutative associative m-ary addition and n-ary associative multiplication
connected by polyadic distributivity (4.5)–(4.7), ν̃p1q
and µ̄p1q
are unary additive queroperation (4.31) and
multiplicative queroperation (2.13).
There is no initial relation between ν̃p1q
and µ̄p1q
, nevertheless a possible their “interaction” can lead to
further thorough classification of polyadic fields.
Definition 4.13. A polyadic field Fpm,nq
is called quer-symmetric, if its unary queroperations commute
ν̃p1q
˝ µ̄p1q
“ µ̄p1q
˝ ν̃p1q
, (4.33)
r
x “ r
x, @x P R, (4.34)
– 16 –

Polyadic hetero product of pm, nq-fields POLYADIC PRODUCTS OF RINGS AND FIELDS
in other case Fpm,nq
is called quer-nonsymmetric.
Example 4.14. Consider the nonunital zeroless (denoted by hat x
F) polyadic field x
Fp3,3q
“
@
tia{bu | νp3q
, µp3q
D
, i2
“ ´1, a, b P Zodd
. The ternary addition νp3q
rx, y, ts “ x ` y ` t and the
ternary multiplication µp3q
rx, y, ts “ xyt are nonderived, ternary associative and distributive (operations
are in C). For each x “ ia{b (a, b P Zodd
) the additive querelement is x̃ “ ´ia{b, and the multiplicative
querelement is x̄ “ ´ib{a (see (2.12)). Therefore, both
@
tia{bu | µp3q
D
and
@
tia{bu | νp3q
D
are ternary
groups, but they both contain no neutral elements (no unit, no zero).The nonunital zeroless p3, 3q-field
x
Fp3,3q
is quer-symmetric, because (see (4.34))
r
x “ r
x “ i
b
a
. (4.35)
To find quer-nonsymmetric polyadic fields is not a simple task.
Example 4.15. Consider the set of real 4 ˆ 4 matrices over the fractions 4k`3
4l`3
, k, l P Z, of the form
M “
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
0
4k1 ` 3
4l1 ` 3
0 0
0 0
4k2 ` 3
4l2 ` 3
0
0 0 0
4k3 ` 3
4l3 ` 3
4k4 ` 3
4l4 ` 3
0 0 0
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
, ki, li P Z. (4.36)
The set tMu is closed with respect to the ordinary addition of m ě 5 matrices, because the sum of feweer
of the fractions 4k`3
4l`3
does not give a fraction of the same form DUPLIJ [2017], and with respect to the
ordinary multiplication of n ě 5 matrices, since the product of fewer matrices (4.36) does not have the
same shape DUPLIJ [2021]. The polyadic associativity and polyadic distributivity follow from the binary
ones of the ordinary matrices over R, and the product of 5 matrices is semicommutative (see 2.10). Taking
the minimal values m “ 5, n “ 5, we define the semicommutative zeroless p5, 5q-field (see (4.11))
Fp5,5q
M “
@
tMu | νp5q
, µp5q
, ν̃p1q
, µ̄p1q
D
, (4.37)
where νp5q
and µp5q
are the ordinary sum and product of 5 matrices, while ν̃p1q
and µ̄p1q
are additive and
multiplicative queroperations
ν̃p1q
rMs ” M̃ “ ´3M, µ̄p1q
rMs ” M̄ “
4l1 ` 3
4k1 ` 3
4l2 ` 3
4k2 ` 3
4l3 ` 3
4k3 ` 3
4l4 ` 3
4k4 ` 3
M. (4.38)
The division ring Dp5,5q
M is zeroless, because the fraction 4k`3
4l`3
, is never zero for k, l P Z, and it is unital
with the unit
Me “
¨
˚
˚
˝
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
˛
‹
‹
‚. (4.39)
Using (4.36) and (4.38), we obtain
ν̃p1q
“
µ̄p1q
rMs
‰
“ ´3
4l1 ` 3
4k1 ` 3
4l2 ` 3
4k2 ` 3
4l3 ` 3
4k3 ` 3
4l4 ` 3
4k4 ` 3
M, (4.40)
µ̄p1q
“
ν̃p1q
rMs
‰
“ ´
1
27
4l1 ` 3
4k1 ` 3
4l2 ` 3
4k2 ` 3
4l3 ` 3
4k3 ` 3
4l4 ` 3
4k4 ` 3
M, (4.41)
– 17 –

or
Ă
M “ 81Ă
M, (4.42)
and therefore the additive and multiplicative queroperations do not commute independently of the field
parameters. Thus, the matrix p5, 5q-division ring Dp5,5q
M (4.37) is a quer-nonsymmetric division ring.
Definition 4.16. The polyadic zeroless direct product field x
F1pm,nq
“
@
R1
| ν1pmq
, µ1pnq
D
consists of (two)
zeroless polyadic fields x
Fpm,nq
1 “
A
R1 | ν
pmq
1 , µ
pnq
1
E
and x
Fpm,nq
2 “
A
R2 | ν
pmq
2 , µ
pnq
2
E
of the same arity
shape, while the componentwise operations on the doubles X P R1 ˆ R2 in (4.10)–(4.11) still hold valid,
and
A
R1 | µ
pnq
1
E
,
A
R2 | µ
pnq
2
E
,
@
R1
“ tXu | µ1pnq
D
are n-ary groups.
Following Definition 4.11, we have
Corollary 4.17. If at least one of the constituent fields is semicommutative, and another one is totally
commutative, then the polyadic product will be a semicommutative pm, nq-field.
The additive and multiplicative unary queroperations (2.13) and (4.31) for the direct product field
x
Fpm,nq
are defined componentwise on the doubles X as follows
ν̃1p1q
rXs “
˜
ν̃
p1q
1 rx1s
ν̃
p1q
2 rx2s
¸
, (4.43)
µ̄1p1q
rXs “
˜
µ̄
p1q
1 rx1s
µ̄
p1q
2 rx2s
¸
, x1 P R1, x2 P R2. (4.44)
Definition 4.18. A polyadic direct product field x
F1pm,nq
“
@
R1
| ν1pmq
, ν̃1p1q
, µ1pnq
, µ̄1p1q
D
is called quer-
symmetric, if its unary queroperations (4.43)–(4.44) commute
ν̃1p1q
˝ µ̄1p1q
“ µ̄1p1q
˝ ν̃1p1q
, (4.45)
Ă
X “ Ă
X, @X P R1
, (4.46)
in other case x
F1pm,nq
is called a quer-nonsymmetric direct product pm, nq-field.
Example 4.19. Consider two nonunital zeroless p3, 3q-fields x
Fp3,3q
1,2 “
A
tia1,2{b1,2u | ν
p3q
1,2 , µ
p3q
1,2, ν̃
p1q
1,2 , µ̄
p1q
1,2
E
,
i2
“ ´1, a1,2, b1,2 P Zodd
, where ternary additions ν
p3q
1,2 and ternary multiplications µ
p3q
1,2 are sum
and product in Zodd
, correspondingly, and the unary additive and multiplicative queroperations are
ν̃
p1q
1,2 ria1,2{b1,2s “ ´ia1,2{b1,2 and µ̄
p1q
1,2 ria1,2{b1,2s “ ´ib1,2{a1,2 (see Example 4.14). Using (4.10)–(4.11)
we build the operations of the polyadic nonderived nonunital zeroless product p3, 3q-field x
F1p3,3q
“
x
Fp3,3q
1 ˆ x
Fp3,3q
2 on the doubles XT
“ pia1{b1, ia2{b2q as follows
ν1p3q
“
Xp1q
, Xp2q
, Xp3q
‰
“
¨
˚
˚
˚
˝
i
a
p1q
1 b
p2q
1 b
p3q
1 ` b
p1q
1 a
p2q
1 b
p3q
1 ` b
p1q
1 b
p2q
1 a
p3q
1
b
p1q
1 b
p2q
1 b
p3q
1
i
a
p1q
2 b
p2q
2 b
p3q
2 ` b
p1q
2 a
p2q
2 b
p3q
2 ` b
p1q
2 b
p2q
2 a
p3q
2
b
p1q
2 b
p2q
2 b
p3q
2
˛
‹
‹
‹
‚
, (4.47)
– 18 –

Polyadic hetero product of pm, nq-fields POLYADIC PRODUCTS OF RINGS AND FIELDS
µ1p3q
“
Xp1q
, Xp2q
, Xp3q
‰
“
¨
˚
˚
˚
˝
í
a
p1q
1 a
p2q
1 a
p3q
1
b
p1q
1 b
p2q
1 b
p3q
1
í
a
p1q
2 a
p2q
2 a
p3q
2
b
p1q
2 b
p2q
2 b
p3q
2
˛
‹
‹
‹
‚
, a
pjq
i , b
pjq
i P Zodd
, (4.48)
and the unary additive and multiplicative queroperations (4.43)–(4.44) of the direct product x
F1p3,3q
are
ν̃1p1q
rXs “
¨
˚
˝
í
a1
b1
í
a2
b2
˛
‹
‚, (4.49)
µ̄1p1q
rXs “
¨
˚
˚
˝
í
b1
a1
í
b2
a2
˛
‹
‹
‚, ai, bi P Zodd
. (4.50)
Therefore, both
@
tXu | ν1p3q
, ν̃1p1q
D
and
@
tXu | µ1p3q
, µ̄1p1q
D
are commutative ternary groups, which
means that the polyadic direct product x
F1p3,3q
“ x
Fp3,3q
1 ˆ x
Fp3,3q
2 is the nonunital zeroless polyadic field.
Moreover, x
F1p3,3q
is quer-symmetric, because (4.45)–(4.46) hold valid
µ̄1p1q
˝ ν̃1p1q
rXs “ ν̃1p1q
˝ µ̄1p1q
rXs “
¨
˚
˚
˝
i
b1
a1
i
b2
a2
˛
‹
‹
‚, ai, bi P Zodd
. (4.51)
Example 4.20. Let us consider the polyadic direct product of two zeroless fields, one of them the
semicommutative p5, 5q-field x
Fp5,5q
1 “ Fp5,5q
M from (4.37), and the other one the nonderived nonuni-
tal zeroless p5, 5q-field of fractions x
Fp5,5q
2 “
A ?
i4r`1
4s`1
(
| ν
p5q
2 , µ
p5q
2
E
, r, s P Z, i2
“ ´1. The dou-
ble is XT
“
`?
i4r`1
4s`1
, M
˘
, where M is in (4.36). The polyadic nonunital zeroless direct product
field x
F1p5,5q
“ x
Fp5,5q
1 ˆ x
Fp5,5q
2 is nonderived and semicommutative, and is defined by x
Fp5,5q
“
@
X | ν1p5q
, µ1p5q
, ν̃1p1q
, µ̄p1q
D
, where its addition and multiplication are
ν1p5q
“
Xp1q
, Xp2q
, Xp3q
, Xp4q
, Xp5q
‰
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
?
i
4Rν ` 1
4Sν ` 1
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
0
4Kν,1 ` 3
4Lν,1 ` 3
0 0
0 0
4Kν,2 ` 3
4Lν,2 ` 3
0
0 0 0
4Kν,3 ` 3
4Lν,3 ` 3
4Kν,4 ` 3
4Lν,4 ` 3
0 0 0
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
,
(4.52)
– 19 –

µ1p5q
“
Xp1q
, Xp2q
, Xp3q
, Xp4q
, Xp5q
‰
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
?
i
4Rµ ` 1
4Sµ ` 1
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
0
4Kµ,1 ` 3
4Lµ,1 ` 3
0 0
0 0
4Kµ,2 ` 3
4Lµ,2 ` 3
0
0 0 0
4Kµ,3 ` 3
4Lµ,3 ` 3
4Kµ,4 ` 3
4Lµ,4 ` 3
0 0 0
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
,
(4.53)
where Rν,µ, Sν,µ P Z are cumbersome integer functions of rpiq
, spiq
P Z, i “ 1, . . . , 5, and
Kν,i, Kµ,i, Lν,i, Lµ,i P Z are cumbersome integer functions of k
piq
j , l
piq
j P Z, j “ 1, . . . , 4, i “ 1, . . . , 5 (see
(4.36)). The unary queroperations (4.43)–(4.44) of the direct product x
Fp5,5q
are
ν̃1p1q
rXs “
¨
˝
´3
?
i
4r ` 1
4s ` 1
´3M
˛
‚, (4.54)
µ̄1p1q
rXs “
¨
˚
˚
˝
´
?
i
ˆ
4s ` 1
4r ` 1
˙3
4l1 ` 3
4k1 ` 3
4l2 ` 3
4k2 ` 3
4l3 ` 3
4k3 ` 3
4l4 ` 3
4k4 ` 3
M
˛
‹
‹
‚, r, s, ki, li P Z, (4.55)
where M is in (4.36). Therefore,
@
tXu | ν1p5q
, ν̃1p1q
D
is a commutative 5-ary group, and
@
tXu | µ1p5q
, µ̄1p1q
D
is a semicommutative 5-ary group, which means that the polyadic direct product
x
F1p5,5q
“ x
Fp5,5q
1 ˆ x
Fp5,5q
2 is the nonunital zeroless polyadic semicommutative p5, 5q-field. Using (4.42)
we obtain
ν̃1p1q
µ̄1p1q
rXs “ 81µ̄1p1q
ν̃1p1q
rXs , (4.56)
and therefore the direct product p5, 5q-field x
F1p5,5q
is quer-nonsymmetric (see (4.33)).
Thus, we arrive at
Theorem 4.21. The category of zeroless polyadic fields zlessPolField can exist (having the class of
all zeroless polyadic fields for objects and field homomorphisms for morphisms) and can be well-defined,
because it has a product as the polyadic field product.
Further analysis of the direct product constructions introduced here and their examples for polyadic
rings and fields would be interesting to provide in detail, which can also lead to new kinds of categories.
Acknowledgement. The author is deeply grateful to Vladimir Akulov, Mike Hewitt, Vladimir Tkach,
Raimund Vogl and Wend Werner for numerous fruitful discussions, important help and valuable support.
– 20 –

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