S. Duplij. Polyadic algebraic structures and their applications

Polyadic algebraic structures
and their applications
STEVEN DUPLIJ
M¨unster, Germany
http://wwwmath.uni-muenster.de/u/duplij
Chern Institute of Mathematics - 2018
1

1 History
1 History
Ternary algebraic operations (with the arity n = 3) were introduced by A. Cayley
in 1845 and later by J. J. Sylvester in 1883.
The notion of an n-ary group was introduced in 1928 by D ÖRNTE [1929] (inspired
by E. Nöther).
The 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].
The cubic and n-ary generalizations of matrices and determinants were made in
KAPRANOV ET AL. [1994], SOKOLOV [1972], physical application in KAWAMURA
[2003], RAUSCH DE TRAUBENBERG [2008].
2

1 History
Particular questions of ternary group representations were considered in
BOROWIEC, DUDEK, AND DUPLIJ [2006], DUPLIJ [2012].
Theorems connecting representations of binary and n-ary groups were given in
DUDEK AND SHAHRYARI [2012].
Ternary fields were developed in DUPLIJ AND WERNER [2015], DUPLIJ [2017b].
In physics, the most applicable structures are the nonassociative Grassmann,
Clifford and Lie algebras L ˜OHMUS ET AL. [1994], GEORGI [1999]. The ternary
analog of Clifford algebra was considered in ABRAMOV [1995], and the ternary
analog of Grassmann algebra ABRAMOV [1996] was exploited to construct
ternary extensions of supersymmetry ABRAMOV ET AL. [1997].
Then binary Lie bracket was replaced by a n-ary bracket, and the algebraic
structure of physical model was defined by the additional characteristic identity for
this generalized bracket, corresponding to the Jacobi identity DE AZCARRAGA
AND IZQUIERDO [2010].
3

1 History
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].
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].
4

2 Plan
2 Plan
1. Classification of general polyadic systems and special elements.
2. Definition of n-ary semigroups and groups.
3. Homomorphisms of polyadic systems.
4. The Hossz´u-Gluskin theorem and its “q-deformed” generalization.
5. Multiplace generalization of homorphisms - heteromorpisms.
6. Associativity quivers.
7. Multiplace representations and multiactions.
8. Examples of matrix multiplace representations for ternary groups.
9. Polyadic rings and fields
10. Polyadic analogs of the integer number ring Z and the Galois field GF(p).
11. Equal sums of like powers Diophantine equation over polyadic integer numbers
5

3 Notations
3 Notations
Let G be a underlying set, universe, carrier, gi ∈ G.
The n-tuple (or polyad) g1, . . . , gn is denoted by (g1, . . . , gn).
The Cartesian product G×n
consists of all n-tuples (g1, . . . , gn).
For equal elements g ∈ G, we denote n-tuple (polyad) by (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), or g(n)
.
The i-projection Pr
(n)
i : G×n
→ G is (g1, . . . gi, . . . , gn) −→ gi.
The i-diagonal Diagn : G → G×n
sends one element to the equal element
n-tuple g −→ (gn
).
6

3 Notations
The one-point set {•} is a “unit” for the Cartesian product, since there are
bijections between G and G × {•}
×n
, and denote it by .
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 Greek letters and 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, 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 BERGMAN [1995]
μn+m−1 [g, h] = μn [g, μm [h]] . (3.1)
Then, if to compose μn with the 0-ary operation μ
(c)
0 , we obtain
μ
(c)
n−1 [g] = μn [g, c] , (3.2)
7

3 Notations
because g is a polyad of length (n − 1). It is also seen from the commutative
diagram
G×(n−1)
× {•}
id×(n−1)
×μ
(c)
0
G×n
G×(n−1) μ
(c)
n−1
G
μn (3.3)
which is a definition of a new (n − 1)-ary operation μ
(c)
n−1.
Remark 3.1. It is important to make a clear distinction between the 0-ary
operation μ
(c)
0 and its value c in G.
8

4 Polyadic systems
4 Polyadic systems
Definition 4.1. A polyadic system G (polyadic algebraic structure) is a set G
together with polyadic operations, which is closed under them.
Here, we mostly consider concrete polyadic systems with one “chief”
(fundamental) n-ary operation μn, which is called polyadic multiplication (or
n-ary multiplication).
Definition 4.2. A n-ary system Gn = G | μn is a set G closed under one
n-ary operation μn (without any other additional structure).
Let us consider the changing arity problem:
Definition 4.3. For a given n-ary system G | μn to construct another polyadic
system G | μn over the same set G, which has multiplication with a different
arity n .
9

4 Polyadic systems
There are 3 ways to change arity of operation:
1. Iterating. Using composition of the operation μn with itself, one can increase
the arity from n to niter . Denote the number of iterating multiplications by
μ, and use the bold Greek letters μ μ
n for the resulting composition of n-ary
multiplications
μn = μ μ
n
def
=
μ
μn ◦ μn ◦ . . . μn × id×(n−1)
. . . × id×(n−1)
,
(4.1)
where n = niter = μ (n − 1) + 1, which gives the length of a polyad
(g) in the notation μ μ
n [g]. The operation μ μ
n is named a long product
D ¨ORNTE [1929] or derived DUDEK [2007].
10

4 Polyadic systems
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 nred (as in (3.2)), such that
μn = μ
(c1...cnc )
n
def
= μn ◦



nc
μ
(c1)
0 × . . . × μ
(cnc )
0 × id×(n−nc)


 ,
(4.2)
where
n = nred = n − nc, (4.3)
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).
11

5 Special elements and properties of n-ary systems
Definition 5.1. A zero
μn [g, z] = z, (5.1)
where z can be on any place in the l.h.s. of (5.1).
Only one zero (if its place is not fixed) can be possible in a polyadic system.
An analog of positive powers of an element POST [1940] should coincide with the
number of multiplications μ in the iterating (4.1).
Definition 5.2. A (positive) polyadic power of an element is
g μ
= μ μ
n g μ(n−1)+1
. (5.2)
12

Example 5.3. Consider a polyadic version of the binary q-addition which appears
in study of nonextensive statistics (see, e.g., TSALLIS [1994], NIVANEN ET AL.
[2003])
μn [g] =
n
i=1
gi +
n
i=1
gi, (5.3)
where gi ∈ C and = 1 − q0, q0 is a real constant ( q0 = 1 or = 0). It is
obvious that g 0
= g, and
g 1
= μn gn−1
, g 0
= ng + gn
, (5.4)
g k
= μn gn−1
, g k−1
= (n − 1) g + 1 + gn−1
g k−1
. (5.5)
Solving this recurrence formula for we get
g k
= g 1 +
n − 1
g1−n
1 + gn−1 k
−
n − 1
g2−n
. (5.6)
13

Definition 5.4. An element of a polyadic system g is called μ-nilpotent (or
simply nilpotent for μ = 1), if there exist such μ that
g μ
= z. (5.7)
Definition 5.5. A polyadic system with zero z is called μ-nilpotent, if there exists
μ such that for any ( μ (n − 1) + 1)-tuple (polyad) g we have
μ μ
n [g] = z. (5.8)
Therefore, the index of nilpotency (number of elements whose product is zero) of
an μ-nilpotent n-ary system is ( μ (n − 1) + 1), while its polyadic power is μ .
14

Definition 5.6. A polyadic (n-ary) identity (or neutral element) of a polyadic
system is a distinguished element ε (and the corresponding 0-ary operation μ
(ε)
0 )
such that for any element g ∈ G we have ROBINSON [1958]
μn g, εn−1
= g, (5.9)
where g can be on any place in the l.h.s. of (5.9).
In binary groups the identity is the only neutral element, while in polyadic systems,
there exist many neutral polyads n consisting of elements of G satisfying
μn [g, n] = g, (5.10)
where g can be also on any place. The neutral polyads are not determined
uniquely.
The sequence of polyadic identities εn−1
is a neutral polyad.
15

Definition 5.7. An element of a polyadic system g is called μ-idempotent (or
simply idempotent for μ = 1), if there exist such μ that
g μ
= g. (5.11)
Both zero and the identity are μ-idempotents with arbitrary μ.
We define (total) associativity as invariance of the composition of two n-ary
multiplications
μ2
n [g, h, u] = μn [g, μn [h] , u] = inv. (5.12)
Informally, “internal brackets/multiplication can be moved on any place”, which
gives n relations
μn ◦ μn × id×(n−1)
= . . . = μn ◦ id×(n−1)
×μn . (5.13)
There are many other particular kinds of associativity THURSTON [1949] and
studied in BELOUSOV [1972], SOKHATSKY [1997].
Definition 5.8. A polyadic semigroup (n-ary semigroup) is a n-ary system in
which the operation is associative, or Gsemigrp
n = G | μn | associativity .
16

In a polyadic system with zero (5.1) one can have trivial associativity, when all n
terms are (5.12) are equal to zero, i.e.
μ2
n [g] = z (5.14)
for any (2n − 1)-tuple g.
Proposition 5.9. Any 2-nilpotent n-ary system (having index of nilpotency
(2n − 1)) is a polyadic semigroup.
17

It is very important to find the associativity preserving conditions, where an
associative initial operation μn leads to an associative final operation μn during
the change of arity.
Example 5.10. 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] . (5.15)
Associativity preserving mixing constructions with different arities and places
were considered in DUDEK AND MICHALSKI [1984], MICHALSKI [1981],
SOKHATSKY [1997].
Definition 5.11. A totally associative polyadic system with identity ε, satisfying
(5.9) μn g, εn−1
= g is called a polyadic monoid.
The structure of any polyadic monoid is fixed POP AND POP [2004]: iterating a
binary operation ˇCUPONA AND TRPENOVSKI [1961].
Several analogs of binary commutativity of polyadic system.
18

A polyadic system is σ-commutative, if μn = μn ◦ σ
μn [g] = μn [σ ◦ g] , (5.16)
where σ ◦ g = gσ(1), . . . , gσ(n) is a permutated polyad and σ is a fixed
element of Sn. If (5.16) 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] , (5.17)
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 . (5.18)
Therefore: if a n-ary semigroup Gsemigrp
is iterated from a commutative binary
semigroup with identity, then Gsemigrp
is semicommutative.
19

Another way to generalize commutativity to polyadic case is to generalize
mediality. In semigroups the binary mediality is
(g11 ∙ g12) ∙ (g21 ∙ g22) = (g11 ∙ g21) ∙ (g12 ∙ g22) , (5.19)
and follows from binary commutativity. In polyadic (n-ary) case they are different.
Definition 5.12. A polyadic system is medial (entropic), if
( EVANS [1963], BELOUSOV [1972])
μn





μn [g11, . . . , g1n]
...
μn [gn1, . . . , gnn]





= μn





μn [g11, . . . , gn1]
...
μn [g1n, . . . , gnn]





. (5.20)
The semicommutative polyadic semigroups are medial, as in the binary case, but,
in general (except n = 3) not vice versa GŁAZEK AND GLEICHGEWICHT [1982].
20

Definition 5.13. A polyadic system is cancellative, if
μn [g, t] = μn [h, t] =⇒ g = h, (5.21)
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.
Definition 5.14. 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 (5.22)
where h can be on any i-th place.
Definition 5.15. A polyadic system which is uniquely i-solvable for all places i is
called a n-ary (or polyadic) quasigroup.
Definition 5.16. An associative polyadic quasigroup is called a n-ary (or
polyadic) group.
21

In a polyadic group the only solution of (5.22) μn [u, h, t] = g is called a
querelement of g and denoted by ˉg D ÖRNTE [1929]
μn [h, ˉg] = g, (5.23)
where ˉg can be on any place. Any idempotent g coincides with its querelement
ˉg = g. It follows from (5.23) and (5.10), that the polyad
ng = gn−2
ˉg (5.24)
is neutral for any element of a polyadic group, where ˉg can be on any place. The
number of relations in (5.23) 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 (5.25)
hold true for any allowable i, j. Analog of g ∙ h ∙ h−1
= h ∙ h−1
∙ g = g.
22

The relation (5.23) can be treated as a definition of the unary queroperation
ˉμ1 [g] = ˉg. (5.26)
Definition 5.17. A polyadic group is a universal algebra
G
grp
n = G | μn, ˉμ1 | associativity, D¨ornte relations , (5.27)
where μn is n-ary associative operation and ˉμ1 is the queroperation (5.26), such
that the following diagram
G×(n) id×(n−1)
×ˉμ1
G×n ˉμ1×id×(n−1)
G×n
G × G
id ×Diag(n−1)
Pr1
G
μn
Pr2
G × G
Diag(n−1)×id
(5.28)
commutes, where ˉμ1 can be only on the first and second places from the right
(resp. left) on the left (resp. right) part of the diagram.
23

A straightforward generalization of the queroperation concept and corresponding
definitions can be made by substituting in the above formulas (5.23)–(5.26) the
n-ary multiplication μn by the iterating multiplication μ μ
n (4.1) (cf. DUDEK [1980]
for μ = 2 and GAL’MAK [2007]).
Definition 5.18. Let us define the querpower k of g recursively
ˉg k
= ˉg k−1 , (5.29)
where ˉg 0
= g, ˉg 1
= ˉg, or as the k composition
ˉμ◦k
1 =
k
ˉμ1 ◦ ˉμ1 ◦ . . . ◦ ˉμ1 of the queroperation (5.26).
For instance, ˉμ◦2
1 = μn−3
n , such that for any ternary group ˉμ◦2
1 = id, i.e. one
has ˉg = g.
24

The negative polyadic power of an element g by (after use of (5.2))
μn g μ−1
, gn−2
, g − μ
= g, μ μ
n g μ(n−1)
, g − μ
= g. (5.30)
Connection of the querpower and the polyadic power by the Heine numbers
HEINE [1878] or q-numbers KAC AND CHEUNG [2002]
[[k]]q =
qk
− 1
q − 1
, (5.31)
which have the “nondeformed” limit q → 1 as [k]q → k. Then
ˉg k
= g −[[k]]2−n , (5.32)
Assertion 5.19. The querpower coincides with the negative polyadic deformed
power with the “deformation” parameter q which is equal to the “deviation”
(2 − n) from the binary group.
25

6 (One-place) homomorphisms of polyadic systems
Let Gn = G | μn and Gn = G | μn be two polyadic systems of any kind
(quasigroup, semigroup, group, etc.). If they have the multiplications of the same
arity n = n , then one can define the mappings from Gn to Gn. Usually such
polyadic systems are similar, and we call mappings between them the equiary
mappings.
Let us take n + 1 (one-place) mappings ϕGG
i : G → G , i = 1, . . . , n + 1.
An ordered system of mappings ϕGG
i is called a homotopy from Gn to Gn, if
ϕGG
n+1 (μn [g1, . . . , gn]) = μn ϕGG
1 (g1) , . . . , ϕGG
n (gn) , gi ∈ G.
(6.1)
26

In general, one should add to this definition the “mapping” of the multiplications
μn
ψ
(μμ )
nn
→ μn . (6.2)
In such a way, the 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
μn (6.3)
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 = μn, then the set ϕGG
i ; id is
called an autotopy of the polyadic system Gn.
27

The diagonal counterparts of homotopy, isotopy and autotopy (when all mappings
ϕGG
i coincide) are homomorphism, isomorphism and automorphism.
A homomorphism from Gn to Gn is given, if there exists one mapping
ϕGG
: G → G satisfying
ϕGG
(μn [g1, . . . , gn]) = μn ϕGG
(g1) , . . . , ϕGG
(gn) , gi ∈ G.
(6.4)
Usually the homomorphism is denoted by the same one letter ϕGG
or the
extended pair of mappings ϕGG
; ψ
(μμ )
nn .
They “...are so well known that we shall not bother to define them carefully”
HOBBY AND MCKENZIE [1988].
28

7 Standard Hossz´u-Gluskin theorem
Consider concrete forms of polyadic multiplication in terms of lesser arity
operations.
History. Simplest way of constructing a n-ary product μn from the binary one
μ2 = (∗) is μ = n iteration (4.1) SUSCHKEWITSCH [1935], MILLER [1935]
μn [g] = g1 ∗ g2 ∗ . . . ∗ gn, gi ∈ G. (7.1)
In D ¨ORNTE [1929] it was noted that not all n-ary groups have a product of this
special form.
The binary group G∗
2 = G | μ2 = ∗, e was called a covering group of the
n-ary group Gn = G | μn in POST [1940] (also, TVERMOES [1953]), where a
theorem establishing a more general (than (7.1)) structure of μn [g] in terms of
subgroup structure was given.
29

A manifest form of the n-ary group product μn [g] in terms of the binary one and
a special mapping was found in HOSSZ ´U [1963], GLUSKIN [1965] and is called
the Hossz´u-Gluskin theorem, despite the same formulas having appeared much
earlier in TURING [1938], POST [1940] (relationship between all the formulations
in GAL’MAK AND VOROBIEV [2013]).
Rewrite (7.1) in its equivalent form
μn [g] = g1 ∗ g2 ∗ . . . ∗ gn ∗ e, gi, e ∈ G, (7.2)
where e is a distinguished element of the binary group G | ∗, e , that is the
identity. Now we apply to (7.2) an “extended” version of the homotopy relation
(6.1) with Φi = ψi, i = 1, . . . n, and the l.h.s. mapping Φn+1 = id, but add an
action ψn+1 on the identity e
μn [g] = μ(e)
n [g] = ψ1 (g1) ∗ ψ2 (g2) ∗ . . . ∗ ψn (gn) ∗ ψn+1 (e) . (7.3)
30

The most general form of polyadic multiplication in terms of (n + 1) “extended”
homotopy maps ψi, i = 1, . . . n + 1, the diagram
G×(n)
× {•}
id×n
×μ
(e)
0
G×(n+1) ψ1×...×ψn+1
G×(n+1)
G×(n) μ(e)
n
G
μ×n
2 (7.4)
commutes.
We can correspondingly classify polyadic systems as:
1) Homotopic polyadic systems presented in the form (7.3). (7.5)
2) Nonhomotopic polyadic systems of other than (7.3) form. (7.6)
If the second class is nonempty, it would be interesting to find examples of
nonhomotopic polyadic systems.
31

The main idea in constructing the “automatically” associative n-ary operation μn
in (7.3) is to express the binary multiplication (∗) and the “extended” homotopy
maps ψi in terms of μn itself SOKOLOV [1976]. A simplest binary multiplication
which can be built from μn is (recall (5.15) μ
(c)
2 [g, h] = μn [g, c, h])
g ∗t h = μn [g, t, h] , (7.7)
where t is any fixed polyad of length (n − 2). The equations for the identity e in
a binary group g ∗t e = g, e ∗t h = h, correspond to
μn [g, t, e] = g, μn [e, t, h] = h. (7.8)
We observe from (7.8) that (t, e) and (e, t) are neutral sequences of length
(n − 1), and therefore we take t as a polyadic inverse of e (the identity of the
binary group) considered as an element (but not an identity) of the polyadic
system G | μn , so formally t = e−1
.
32

Then, the binary multiplication is
g ∗ h = g ∗e h = μn g, e−1
, h . (7.9)
Remark 7.1. Using this construction any element of the polyadic system
G | μn can be distinguished and may serve as the identity of the binary group,
and is then denoted by e .
Recognize in (7.9) a version of the Maltsev term (see, e.g., BERGMAN [2012]),
which can be called a polyadic Maltsev term and is defined as
p (g, e, h)
def
= μn g, e−1
, h (7.10)
having the standard term properties p (g, e, e) = g, p (e, e, h) = h.
For n-ary group we can write g−1
= gn−3
, ˉg and the binary group inverse
g−1
is g−1
= μn e, gn−3
, ˉg, e , the polyadic Maltsev term becomes
SHCHUCHKIN [2003]
p (g, e, h) = μn g, en−3
, ˉe, h . (7.11)
33

Derive the Hossz´u-Gluskin “chain formula” for ternary n = 3 case, and then it will
be clear how to proceed for generic n. We write
μ3 [g, h, u] = ψ1 (g) ∗ ψ2 (h) ∗ ψ3 (u) ∗ ψ4 (e) (7.12)
and try to construct ψi in terms of the ternary product μ3 and the binary identity
e. A neutral ternary polyad (ˉe, e) or its powers ˉek
, ek
. Thus, taking for all
insertions the minimal number of neutral polyads, we get
μ3 [g, h, u] = μ7
3









g,
∗
↓
ˉe , e, h, ˉe,
∗
↓
ˉe , e, e, u, ˉe, ˉe,
∗
↓
ˉe , e, e, e









. (7.13)
34

We rewrite (7.13) as
μ3 [g, h, u] = μ3
3









g,
∗
↓
ˉe , μ3 [e, h, ˉe] ,
∗
↓
ˉe , μ2
3 [e, e, u, ˉe, ˉe] ,
∗
↓
ˉe , μ3 [e, e, e]









.
(7.14)
Comparing this with (7.12), we can identify
ψ1 (g) = g, (7.15)
ψ2 (g) = ϕ (g) , (7.16)
ψ3 (g) = ϕ (ϕ (g)) = ϕ2
(g) , (7.17)
ψ4 (e) = μ3 [e, e, e] = e 1
, (7.18)
ϕ (g) = μ3 [e, g, ˉe] . (7.19)
35

Thus, we get the Hossz´u-Gluskin “chain formula” for n = 3
μ3 [g, h, u] = g ∗ ϕ (h) ∗ ϕ2
(u) ∗ b, (7.20)
b = e 1
. (7.21)
The polyadic power e 1
is a fixed point, because ϕ e 1
= e 1
, as well as
higher polyadic powers e k
= μk
3 e2k+1
of the binary identity e are obviously
also fixed points ϕ e k
= e k
.
By analogy, the Hossz´u-Gluskin “chain formula” for arbitrary n can be obtained
using substitution ˉe → e−1
, neutral polyads e−1
, e and their powers
e−1 k
, ek
, the mapping ϕ in the n-ary case is
ϕ (g) = μn e, g, e−1
, (7.22)
and μn [e, . . . , e] is also the first n-ary power e 1
(5.2).
36

In this way, we obtain the Hosszú-Gluskin “chain formula” for arbitrary n
μn [g1, . . . , gn] = g1∗ϕ (g2)∗ϕ2
(g3)∗. . .∗ϕn−2
(gn−1)∗ϕn−1
(gn)∗e 1
.
(7.23)
Thus, we have found the “extended” homotopy maps ψi from (7.3)
ψi (g) = ϕi−1
(g) , i = 1, . . . , n, (7.24)
ψn+1 (g) = g 1
, (7.25)
where by definition ϕ0
(g) = g. Using (7) and (7.23) we can formulate the
standard Hosszú-Gluskin theorem in the language of polyadic powers.
Theorem 7.2. On a polyadic group Gn = G | μn, ˉμ1 one can define a binary
group G∗
2 = G | μ2 = ∗, e and its automorphism ϕ such that the
Hosszú-Gluskin “chain formula” (7.23) is valid.
37

The following reverse Hossz´u-Gluskin theorem holds.
Theorem 7.3. If in a binary group G∗
2 = G | μ2 = ∗, e one can define an
automorphism ϕ such that
ϕn−1
(g) = b ∗ g ∗ b−1
, (7.26)
ϕ (b) = b, (7.27)
where b ∈ G is a distinguished element, then the “chain formula”
μn [g1, . . . , gn] = g1 ∗ϕ (g2)∗ϕ2
(g3)∗. . .∗ϕn−2
(gn−1)∗ϕn−1
(gn)∗b.
(7.28)
determines a n-ary group, in which the distinguished element is the first polyadic
power of the binary identity b = e 1
.
38

8 “Deformation” of Hossz´u-Gluskin chain formula
Idea: to generalize the Hossz´u-Gluskin chain formula DUPLIJ [2016]. We take the
number of the inserted neutral polyads arbitrarily, not only minimally, as they are
all neutral. Indeed, in the particular case n = 3, we put the map ϕ as
ϕq (g) = μ ϕ(q)
3 [e, g, ˉeq
] , (8.1)
where the number of multiplications
ϕ (q) =
q + 1
2
(8.2)
is an integer ϕ (q) = 1, 2, 3 . . ., while q = 1, 3, 5, 7 . . .. Then, we get
μ3 [g, h, u] = μ•
3 g, ˉe, (e, h, ˉeq
) , ˉe, (e, u, ˉeq
)
q+1
, ˉe, eq(q+1)+1
. (8.3)
39

Therefore we have obtained the “q-deformed” homotopy maps
ψ1 (g) = ϕ
[[0]]q
q (g) = ϕ0
q (g) = g, (8.4)
ψ2 (g) = ϕq (g) = ϕ
[[1]]q
q (g) , (8.5)
ψ3 (g) = ϕq+1
q (g) = ϕ
[[2]]q
q (g) , (8.6)
ψ4 (g) = μ•
3 gq(q+1)+1
= μ•
3 g[[3]]q , (8.7)
where ϕ is defined by (8.1) and [[k]]q is the q-deformed number and we put
ϕ0
q = id. The corresponding “q-deformed” chain formula (for n = 3) can be
written as
μ3 [g, h, u] = g ∗ ϕ
[[1]]q
q (h) ∗ ϕ
[[2]]q
q (u) ∗ bq, (8.8)
bq = e e(q)
, (8.9)
e (q) = q
[[2]]q
2
. (8.10)
40

The “nondeformed” limit q → 1 of (8.8) gives the standard Hossz´u-Gluskin chain
formula (7.20) for n = 3. For arbitrary n we insert all possible powers of neutral
polyads e−1 k
, ek
(they are allowed by the chain properties), and obtain
ϕq (g) = μ ϕ(q)
n e, g, e−1 q
, (8.11)
where the number of multiplications ϕ (q) =
q (n − 2) + 1
n − 1
is an integer and
ϕ (q) → q, as n → ∞, and ϕ (1) = 1, as in (7.22).
41

The “deformed” map ϕq is a kind of a-quasi-endomorphism GLUSKIN AND
SHVARTS [1972] (which has one multiplication and leads to the standard
“nondeformed” chain formula) of the binary group G∗
2, because from (8.11) we get
ϕq (g) ∗ ϕq (h) = ϕq (g ∗ a ∗ h) , (8.12)
where a = ϕq (e). A general quasi-endomorphism DUPLIJ [2016]
ϕq (g) ∗ ϕq (h) = ϕq g ∗ ϕq (e) ∗ h . (8.13)
The corresponding diagram
G × G
μ2
G
ϕq
G
G × G
ϕq×ϕq
G × {•} × G
id ×μ
(e)
0 ×id
G × G × G
μ2×μ2 (8.14)
commutes. If q = 1, then ϕq (e) = e, and the distinguished element a turns to
binary identity a = e, and ϕq is an automorphism of G∗
2.
42

The “extended” homotopy maps ψi (7.3) now are
ψ1 (g) = ϕ
[[0]]q
q (g) = ϕ0
q (g) = g, (8.15)
ψ2 (g) = ϕq (g) = ϕ
[[1]]q
q (g) , (8.16)
ψ3 (g) = ϕq+1
q (g) = ϕ
[[2]]q
q (g) , (8.17)
...
ψn−1 (g) = ϕqn−3
+...+q+1
q (g) = ϕ
[[n−2]]q
q (g) , (8.18)
ψn (g) = ϕqn−2
+...+q+1
q (g) = ϕ
[[n−1]]q
q (g) , (8.19)
ψn+1 (g) = μ•
n gqn−1
+...+q+1
= μ•
n g[[n]]q . (8.20)
In terms of the polyadic power (5.2), the last map is
ψn+1 (g) = g e
, e (q) = q
[[n − 1]]q
n − 1
. (8.21)
43

Thus the “q-deformed” n-ary chain formula is DUPLIJ [2016]
μn [g1, . . . , gn] = g1 ∗ ϕ
[[1]]q
q (g2) ∗ ϕ
[[2]]q
q (g3) ∗ . . .
∗ ϕ
[[n−2]]q
q (gn−1) ∗ ϕ
[[n−1]]q
q (gn) ∗ e e(q)
. (8.22)
In the “nondeformed” limit q → 1 (8.22) reproduces the standard Hossz´u-Gluskin
chain formula (7.23).
Instead of the fixed point relation (7.27) ϕ (b) = b we now have the quasi-fixed
point
ϕq (bq) = bq ∗ ϕq (e) , (8.23)
bq = μ•
n e[[n]]q = e e(q)
. (8.24)
The conjugation relation (7.26) ϕn−1
(g) = b ∗ g ∗ b−1
in the “deformed” case
becomes the quasi-conjugation DUPLIJ [2016]
ϕ
[[n−1]]q
q (g) ∗ bq = bq ∗ ϕ
[[n−1]]q
q (e) ∗ g. (8.25)
44

We formulate the following “q-deformed” analog of the Hossz´u-Gluskin theorem
DUPLIJ [2016].
Theorem 8.1. On a polyadic group Gn = G | μn, ˉμ1 one can define a binary
group G∗
2 = G | μ2 = ∗, e and (the infinite “q-series” of) its automorphism ϕq
such that the “deformed” chain formula (8.22) is valid
μn [g1, . . . , gn] = ∗
n
i=1
ϕ[[i−1]]q (gi) ∗ bq, (8.26)
where (the infinite “q-series” of) the “deformed” distinguished element bq (being a
polyadic power of the binary identity (8.24)) is the quasi-fixed point of ϕq (8.23)
and satisfies the quasi-conjugation (8.25) in the form
ϕ
[[n−1]]q
q (g) = bq ∗ ϕ
[[n−1]]q
q (e) ∗ g ∗ b−1
q . (8.27)
45

9 (One-place) generalizations of homomorphisms
Definition 9.1. The n-ary homomorphism is realized as a sequence of n
consequent (binary) homomorphisms ϕi, i = 1, . . . , n, of n similar polyadic
systems
n
Gn
ϕ1
→ Gn
ϕ2
→ . . .
ϕn−1
→ Gn
ϕn
→ Gn (9.1)
Generalized POST [1940] n-adic substitutions in GAL’MAK [1998].
There are two possibilities to change arity:
1) add another equiary diagram with additional operations using the same formula
(6.4), where both do not change arity (are equiary);
2) use one modified (and not equiary) diagram and the underlying formula (6.4)
by themselves, which will allow us to change arity without introducing additional
operations.
46

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 algebras 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].
Incorporate into the polyadic systems G | μn and G | μn the following
additional term operations of opposite arity νn : G×n
→ G and
νn : G ×n
→ G and consider two equiary mappings between G | μn, νn
and G | μn , νn .
47

A weak homomorphism from G | μn, νn to G | μn , νn is given, if there
exists a mapping ϕGG
: G → G satisfying two relations simultaneously
ϕGG
(μn [g1, . . . , gn]) = νn ϕGG
(g1) , . . . , ϕGG
(gn) , (9.2)
ϕGG
(νn [g1, . . . , gn ]) = μn ϕGG
(g1) , . . . , ϕGG
(gn ) . (9.3)
G
ϕGG
G
..........ψ
(μν )
nn
............
G×n
μn
ϕGG
×n
(G )
×n
νn
G
ϕGG
G
............ψ
(νμ )
n n
.............
G×n
νn
ϕGG
×n
(G )
×n
μn (9.4)
If only one of the relations (9.2) or (9.3) holds, such a mapping is called a
semi-weak homomorphism KOLIBIAR [1984]. If ϕGG
is bijective, then it defines
a weak isomorphism.
48

10 Multiplace mappings and heteromorphisms
Second way of changing the arity: use only one relation (diagram). Idea. Using
the additional distinguished mapping: the identity idG. Define an ( id-intact)
id-product for the n-ary system G | μn as
μ( id)
n = μn × (idG)
× id
, (10.1)
μ( id)
n : G×(n+ id)
→ G×(1+ id)
. (10.2)
To indicate the exact i-th place of μn in (10.1), we write μ
( id)
n (i).
Introduce a multiplace mapping Φ
(n,n )
k acting as DUPLIJ [2012]
Φ
(n,n )
k : G×k
→ G . (10.3)
49

We have the following commutative diagram which changes arity
G×k Φk
G
G×kn
μ
( id)
n
(Φk)×n
(G )
×n
μn
(10.4)
Definition 10.1. A k-place heteromorphism from Gn to Gn is given, if there
exists a k-place mapping Φ
(n,n )
k (10.3) such that the corresponding defining
equation (a modification of (6.4)) depends on the place i of μn in (10.1).
50

For i = 1 it can read as DUPLIJ [2012]
Φ
(n,n )
k







μn [g1, . . . , gn]
gn+1
...
gn+ id







= μn





Φ
(n,n )
k





g1
...
gk





, . . . , Φ
(n,n )
k





gk(n −1)
...
gkn










.
(10.5)
In the particular case n = 3, n = 2, k = 2, id = 1 we have
Φ
(3,2)
2


μ3 [g1, g2, g3]
g4

 = μ2

Φ
(3,2)
2


g1
g2

 , Φ
(3,2)
2


g3
g4



 .
(10.6)
This was used in the construction of the bi-element representations of ternary
groups BOROWIEC, DUDEK, AND DUPLIJ [2006], DUPLIJ [2012].
51

Example 10.2. 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. 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

 = a1 a2b1 b2, (10.7)
which satisfies (10.6). Introduce a 1-place mapping by ϕ (gi) = aibi, which
satisfies the standard (6.4) for a commutative field K only (= R, C) becoming a
homomorphism. The relation between the heteromorhism Φ
(3,2)
2 and ϕ
Φ
(3,2)
2


g1
g2

 = ϕ (g1) ∙ ϕ (g2) = a1 b1a2 b2, (10.8)
where the product (∙) is in K, such that (6.4) and (10.6) coincide. For the
noncommutative field K (= Q or H) we can define only the heteromorphism.
52

A heteromorphism is called derived, if it can be expressed through an ordinary
(one-place) homomorphism (as e.g., (10.8)).
A heteromorphism is called a μ-ple heteromorphism, if it contains μ
multiplications in the argument of Φ
(n,n )
k in its defining relation. We define a
μ-ple id-intact id-product for G; μn as
μ( μ, id)
n = (μn)
× μ
× (idG)
× id
, (10.9)
μ( μ, id)
n : G×(n μ+ id)
→ G×( μ+ id)
. (10.10)
A μ-ple k-place heteromorphism from Gn to Gn is given, if there exists a
k-place mapping Φ
(n,n )
k (10.3).
53

The main heteromorphism equation is DUPLIJ [2012]
Φ
n,n
k




















μn [g1, . . . , gn] ,
.
.
.
μn g
n μ−1
, . . . , gn μ



μ
gn μ+1,
.
.
.
gn μ+ id



id




















= μ
n







Φ
n,n
k







g1
.
.
.
gk







, . . . , Φ
n,n
k







g
k n −1
.
.
.
g
kn














.
(10.11)
It is a polyadic analog of binary Φ (g1 ∗ g2) = Φ (g1) • Φ (g2), which
corresponds to n = 2, n = 2, k = 1, μ = 1, id = 0, μ2 = ∗, μ2 = •.
We obtain two arity changing formulas
n = n −
n − 1
k
id, (10.12)
n =
n − 1
k
μ + 1, (10.13)
where n−1
k id ≥ 1 and n−1
k μ ≥ 1 are integer.
54

The following inequalities hold valid
1 ≤ μ ≤ k, (10.14)
0 ≤ id ≤ k − 1, (10.15)
μ ≤ k ≤ (n − 1) μ, (10.16)
2 ≤ n ≤ n. (10.17)
The main statement follows from (10.17):
The heteromorphism Φ
(n,n )
k decreases arity of the multiplication.
If id = 0 then it is change of the arity n = n.
If id = 0, then k = kmin = μ, and no change of arity nmax = n.
We call a heteromorphism having id = 0 a k-place homomorphism with
k = μ. An opposite extreme case, when the final arity approaches its minimum
nmin = 2 (the final operation is binary), corresponds to the maximal value of
places k = kmax = (n − 1) μ.
55

Figure 1:
Dependence of the final arity n through the number of heteromorphism places k
for the fixed initial arity n = 9 with
left: fixed intact elements id = const ( id = 1 (solid), id = 2 (dash));
right: fixed multiplications μ = const ( μ = 1 (solid), μ = 2 (dash)).
56

Theorem 10.3. Any n-ary system can be mapped into a binary system by
binarizing heteromorphism Φ
(n,2)
(n−1) μ
, id = (n − 2) μ.
Proposition 10.4. Classification of μ-ple heteromorphisms:
1. n = nmax = n =⇒ Φ
(n,n)
μ
is the μ-place homomorphism,
k = kmin = μ.
2. 2 < n < n =⇒ Φ
(n,n )
k is the intermediate heteromorphism with
k =
n − 1
n − 1
μ, id =
n − n
n − 1
μ. (10.18)
3. n = nmin = 2 =⇒ Φ
(n,2)
(n−1) μ
is the (n − 1) μ-place binarizing
heteromorphism, i.e., k = kmax = (n − 1) μ.
57

Table 1: “Quantization” of heteromorphisms
k μ id 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, . . .58

11 Associativity, quivers and heteromorphisms
Semigroup heteromorphisms: associativity of the final operation μn , when the
initial operation μn is associative.
A polyadic quiver of product is the set of elements from Gn and arrows, such that
the elements along arrows form n-ary product μn DUPLIJ [2012]. For instance,
for the multiplication μ4 [g1, h2, g2, u1] the 4-adic quiver is denoted by
{g1 → h2 → g2 → u1}.
Define polyadic quivers which are related to the main heteromorphism equation
(10.11).
59

For example, the polyadic quiver {g1 → h2 → g2 → u1; h1, u2} corresponds
to the heteromorphism with n = 4, n = 2, k = 3, id = 2 and μ = 1 such
that
Φ
(4,2)
3




μ4 [g1, h2, g2, u1]
h1
u2



 = μ2



Φ
(4,2)
3




g1
h1
u1



 , Φ
(4,2)
3




g2
h2
u2







 .
(11.1)
As it is seen from here (11.1), the product μ2 is not associative, even if μ4 is
associative.
Definition 11.1. An associative polyadic quiver is a polyadic quiver which
ensures the final associativity of μn in the main heteromorphism equation
(10.11), when the initial multiplication μn is associative.
60

One of the associative polyadic quivers which corresponds to the same
heteromorphism parameters as the non-associative quiver (11.1) is
{g1 → h2 → u1 → g2; h1, u2} which corresponds to
g1 h1 u1 g1 h1 u1
g2 h2 u2 g2 h2 u2
corr
Φ
(4,2)
3




μ4 [g1, h2, u1, g2]
h1
u2



 = μ2



Φ
(4,2)
3




g1
h1
u1



 , Φ
(4,2)
3




g2
h2
u2







 .
(11.2)
We propose a classification of associative polyadic quivers and the rules of
construction of corresponding heteromorphism equations, i.e. consistent
procedure for building semigroup heteromorphisms.
61

The first class of heteromorphisms ( id = 0 or intactless), that is μ-place
(multiplace) homomorphisms. As an example, for n = n = 3, k = 2, μ = 2
we have
Φ
(3,3)
2


μ3 [g1, g2, g3]
μ3 [h1, h2, h3]

 = μ3

Φ
(3,3)
2


g1
h1

 , Φ
(3,3)
2


g2
h2

 , Φ
(3,3)
2


g3
h3




(11.3)
Note that the analogous quiver with opposite arrow directions is
Φ
(3,3)
2


μ3 [g1, g2, g3]
μ3 [h3, h2, h1]

 = μ3

Φ
(3,3)
2


g1
h1

 , Φ
(3,3)
2


g2
h2

 , Φ
(3,3)
2


g3
h3




(11.4)
It was used in constructing the middle representations of ternary groups
BOROWIEC, DUDEK, AND DUPLIJ [2006], DUPLIJ [2012].
62

An important class of intactless heteromorphisms (with id = 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 μ = k = n − 1. An
example of the Post-like associative quiver with the same heteromorphisms
parameters as in (11.3)-(11.4) is
Φ
(3,3)
2


μ3 [g1, h2, g3]
μ3 [h1, g2, h3]

 = μ3

Φ
(3,3)
2


g1
h1

 , Φ
(3,3)
2


g2
h2

 , Φ
(3,3)
2


g3
h3




(11.5)
This construction appeared in the study of ternary semigroups of morphisms
CHRONOWSKI [1994]. Its n-ary generalization was used special representations
of n-groups GLEICHGEWICHT ET AL. [1983], WANKE-JAKUBOWSKA AND
WANKE-JERIE [1984] (where the n-group with the multiplication μ2 was called the
diagonal n-group).
63

12 Multiplace representations of polyadic systems
In the heteromorphism equation final multiplication μn is a linear map, which
leads to restrictions on the final polyadic structure Gn .
Let V be a vector space over a field K 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 (one-place) map
Π1 : G2 → End V , such that Π1 is a (one-place) homomorphism
Π1 (μ2 [g, h]) = Π1 (g) ∗ Π1 (h) , (12.1)
where g, h ∈ G and (∗) is the binary multiplication in End V .
64

If G2 is a binary group with the unity e, then we have the additional (normalizing)
condition
Π1 (e) = idV . (12.2)
General idea: to use the heteromorphism equation (10.11) instead of the
standard homomorphism equation (12.1) Π1 (μ2 [g, h]) = Π1 (g) ∗ Π1 (h),
such that the arity of the representation will be different from the arity of the initial
polyadic system, i.e. n = n.
Consider the structure of the final n -ary multiplication μn in (10.11), taking into
account that the final polyadic system Gn should be constructed from the
endomorphisms End V . The most natural and physically applicable way is to
consider the binary End V and to put Gn = dern (End V ), as it was
proposed for the ternary case in BOROWIEC, DUDEK, AND DUPLIJ [2006],
DUPLIJ [2012].
65

In this way Gn becomes a derived n -ary (semi)group of endomorphisms of V
with the multiplication μn : (End V )
×n
→ End V , where
μn [v1, . . . , vn ] = v1 ∗ . . . ∗ vn , vi ∈ End V. (12.3)
Because the multiplication μn (12.3) is derived and is therefore associative,
consider the associative initial polyadic systems.
Let Gn = G | μn be an associative n-ary polyadic system. By analogy with
(10.3), we introduce the following k-place mapping
Π
(n,n )
k : G×k
→ End V. (12.4)
66

A multiplace representation of an associative polyadic system Gn in a vector
space V is given, if there exists a k-place mapping (12.4) Π
(n,n )
k which satisfies
the (associativity preserving) heteromorphism equation (10.11), that is DUPLIJ
[2012]
Π
n,n
k




















μn [g1, . . . , gn] ,
.
.
.
μn g
n μ−1
, . . . , gn μ



μ
gn μ+1,
.
.
.
gn μ+ id



id




















=
n
Π
n,n
k







g1
.
.
.
gk







∗ . . . ∗ Π
n,n
k







g
k n −1
.
.
.
g
kn







,
(12.5)
G×k Πk
End V
G×kn
μ
( μ, id)
n
(Πk)×n
(End V )
×n
(∗)n
(12.6)
where μ
( μ, id)
n is μ-ple id-intact id-product given by (10.9).
67

General classification of multiplace representations can be done by analogy with
that of the heteromorphisms as follows:
1. The hom-like multiplace representation which is a multiplace homomorphism
with n = nmax = n, without intact elements lid = l
(min)
id = 0, and
minimal number of places k = kmin = μ.
2. The intact element multiplace representation which is the intermediate
heteromorphism with 2 < n < n and the number of intact elements is
lid =
n − n
n − 1
μ. (12.7)
3. The binary multiplace representation which is a binarizing heteromorphism
(3) with n = nmin = 2, the maximal number of intact elements
l
(max)
id = (n − 2) μ and maximal number of places
k = kmax = (n − 1) μ. (12.8)
In case of n-ary groups, we need an analog of the “normalizing” relation (12.2)
68

Π1 (e) = idV . If the n-ary group has the unity e, then
Π
(n,n )
k





e
...
e



k





= idV . (12.9)
69

If there is no unity at all, one can “normalize” the multiplace representation, using
analogy with (12.2) in the form
Π1 h−1
∗ h = idV , (12.10)
as follows
Π
(n,n )
k














ˉh
...
ˉh



μ
h
...
h



id














= idV , (12.11)
for all h ∈Gn, where ˉh is the querelement of h which can be on any places in the
l.h.s. of (12.11) due to the D¨ornte identities.
70

A general form of multiplace representations can be found by applying the D¨ornte
identities to each n-ary product in the l.h.s. of (12.5). Then, using (12.11) we
have schematically
Π
(n,n )
k





g1
...
gk





= Π
(n,n )
k














t1
...
t μ
g
...
g



id














, (12.12)
where g is an arbitrary fixed element of the n-ary group and
ta = μn [ga1, . . . , gan−1, ˉg] , a = 1, . . . , μ. (12.13)
71

13 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} . . . .
(13.1)
If the polyadic system is a n-ary group, then, in addition to (13.1), it can be
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.
72

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 ¨OLLER ET AL. [2008]).
Here we introduce the multiplace concept of action for polyadic systems, which is
consistent with heteromorphisms and multiplace representations.
For a polyadic system Gn = G | μn and a set X we introduce an external
polyadic operation
ρk : G×k
× X → X, (13.2)
which is called a (left) k-place action or multiaction. We use the analogy with
multiplication laws of the heteromorphisms (10.11) . and the multiplace
representations (12.5).
73

We propose (schematically) DUPLIJ [2012]
ρ
(n)
k



μn [g1, . . . , gn] ,
.
.
.
μn g
n μ−1
, . . . , gn μ



μ
gn μ+1,
.
.
.
gn μ+ id



id
x



= ρ
(n)
k
n



g1
.
.
.
gk
. . . ρ
(n)
k



g
k n −1
.
.
.
g
kn
x



. . .



.
(13.3)
The connection between all the parameters here is the same as in the arity
changing formulas (10.12)–(10.13). Composition of mappings is associative, and
therefore in concrete cases we can use our associative quiver technique from
Section 11.
74

If Gn is n-ary group, then we should add to (13.3) the “normalizing” relations. If
there is a unity e ∈Gn, then
ρ
(n)
k



e
...
e
x



= x, for all x ∈ X. (13.4)
In terms of the querelement, the normalization has the form
ρ
(n)
k



ˉh
...
ˉh



μ
h
...
h



id
x



= x, for all x ∈ X and for all h ∈ Gn. (13.5)
75

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. (13.6)
If g1, . . . , gk are unique, then ρ
(n)
k is sharply transitive. The subset of X, in
which any points are connected by (13.6) 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).
76

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, (13.7)
which makes G-space into a category (for details, see, e.g., HUSEM ¨OLLER ET AL.
[2008]). In the particular case, when X is a vector space over K, the multiaction
(13.2) can be called a multi-G-module which satisfies (13.4).
77

The additional (linearity) conditions are
ρ
(n)
k



g1
...
gk
ax + by



= aρ
(n)
k



g1
...
gk
x



+bρ
(n)
k



g1
...
gk
y



, (13.8)
where a, b ∈ K. Then, comparing (12.5) and (13.3) 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



. (13.9)
In a similar way, one can generalize to polyadic systems other notions from group
action theory, see e.g. KIRILLOV [1976].
78

14 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 representations 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, 2001]. Now we will show in examples, how the regular multiactions can
arise from i-translations.
79

Example 14.1. 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] . (14.1)
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



. (14.2)
Thus, using the regular 2-action (14.1) we have, in fact, found the associative
quiver corresponding to (10.6).
80

The formula (14.1) 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] , (14.3)
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. In general, the value of i
fixes the minimal final arity nreg, which differs for even and odd values of the
initial arity n.
81

It follows from (14.3) that for regular multiactions the number of places is fixed
kreg = n − 1, (14.4)
and the arity changing formulas (10.12)–(10.13) become
nreg = n − id (14.5)
nreg = μ + 1. (14.6)
From (14.5)–(14.6) we conclude that for any n a regular multiaction having one
multiplication μ = 1 is binarizing and has n − 2 intact elements. For n = 3 see
(14.2). Also, it follows from (14.5) that for regular multiactions the number of intact
elements gives exactly the difference between initial and final arities.
82

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 (11.4) and such 2-actions were used in
BOROWIEC, DUDEK, AND DUPLIJ [2006], DUPLIJ [2012] to construct middle
representations of ternary 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.
83

Example 14.2. 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

 . (14.7)
Using (14.6) we observe that there are two possibilities for the number of
multiplications μ = 2, 4. The last case μ = 4 is similar to the vertical
84

associative quiver (11.4), 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









(14.8)
85

Now we have an additional case with two intact elements id and two
multiplications μ = 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









,
(14.9)
with arity changing from n = 5 to nreg = 3. In addition to (14.9) 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.
86

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 nreg = n. Therefore, the composition of
multiactions will be similar to (14.8), 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 multiactions (14.3) in the r.h.s. of (13.9). 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 ´C AND
ARTAMONOV [1999, 2002]).
87

This means that any element of KGn can be uniquely presented in the form
w = l al ∙ hl, al ∈ K, hl ∈ G. Then, using (14.3), we define the i-regular
k-place representation DUPLIJ [2012]
Π
reg(i)
k





g1
...
gk





(w) =
l
al ∙ μk+1 [g1 . . . gi−1hlgi+1 . . . gk] . (14.10)
Comparing (14.3) and (14.10) one can conclude that 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. 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,
DUDEK, AND DUPLIJ [2006], DUPLIJ [2012] for ternary groups.
88

15 Matrix representations of ternary groups
Here we give several examples of matrix representations for concrete ternary
groups BOROWIEC, DUDEK, AND DUPLIJ [2006], DUPLIJ [2012].
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. (15.1)
The calculations give the left regular representation in the manifest matrix form
Π
L
reg (0, 0) = Π
L
reg (2, 2) = Π
L
reg (1, 1) = Π
R
reg (0, 0)
= Π
R
reg (2, 2) = Π
R
reg (1, 1) =




1 0 0
0 1 0
0 0 1



 = [1] ⊕ [1] ⊕ [1], (15.2)
89

Π
L
reg (2, 0) = Π
L
reg (1, 2) = Π
L
reg (0, 1) = Π
R
reg (2, 1) = Π
R
reg (1, 0) = Π
R
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 , (15.3)
Π
L
reg (2, 1) = Π
L
reg (1, 0) = Π
L
reg (0, 2) = Π
R
reg (2, 0) = Π
R
reg (1, 2) = Π
R
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 . (15.4)
90

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) , (15.5)
ΠM
reg (g1, h1) ◦ ΠL
reg (g2, h2) = ΠL
reg (g1, g2) ◦ ΠM
reg (h2, h1) . (15.6)
91

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
92

sum of two irreducible 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.
93

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. (15.7)
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







94

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 (14.10))
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.
95

In this case the equivalence is
ΠL
(a, b) = ΠL
(c, d) ⇐⇒ (a + b) = (c + d)mod 4, and we obtain the
following classes
Π
L
reg (0, 0) = Π
L
reg (1, 3) = Π
L
reg (2, 2) = Π
L
reg (3, 1) =







0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0







= [1] ⊕ [−1] ⊕ [−i] ⊕ [i] ,
Π
L
reg (0, 1) = Π
L
reg (1, 0) = Π
L
reg (2, 3) = Π
L
reg (3, 2) =







0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0







= [1] ⊕ [−1] ⊕ [−1] ⊕ [−1] ,
Π
L
reg (0, 2) = Π
L
reg (1, 1) = Π
L
reg (2, 0) = Π
L
reg (3, 3) =







0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0







= [1] ⊕ [−1] ⊕ [i] ⊕ [−i] ,
Π
L
reg (0, 3) = Π
L
reg (1, 2) = Π
L
reg (2, 1) = Π
L
reg (3, 0) =







1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1







= [1] ⊕ [−1] ⊕ [1] ⊕ [1] .
Due to the fact that the ternary operation (15.7) is commutative, there are only
one-dimensional irreducible left representations.
96

Let us “algebralize” the regular representations DUPLIJ [2012]. From (7.15) we
have, for the left representation
ΠL
reg (i, j) ◦ ΠL
reg (k, l) = ΠL
reg (i, [jkl]) , (15.8)
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. (15.9)
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. (15.10)
In some sense (15.10) can be treated as a ternary analog of the Clifford algebra.
Now the middle representation is ΠM
(k, l) = γk+l.
97

16 Polyadic rings and fields
Polyadic distributivity
Let us consider a polyadic structure with two operations on the same set R: the
“chief” (multiplication) n-ary operation μn : Rn
→ R and the additional m-ary
operation νm : Rm
→ R, that is R | μn, νm . If there are no relations
between μn and νm, then nothing new, as compared with the polyadic
structures having a single operation R | μn or R | νm , can be said.
Informally, the “interaction” between operations can be described using the
important relation of distributivity (an analog of a ∙ (b + c) = a ∙ b + a ∙ c).
98

Definition 16.1. The polyadic distributivity for the operations μn and νm (no
additional properties are implied for now) consists of n relations
μn [νm [a1, . . . am] , b2, b3, . . . bn]
= νm [μn [a1, b2, b3, . . . bn] , μn [a2, b2, b3, . . . bn] , . . . μn [am, b2, b3, . . . bn]]
(16.1)
μn [b1, νm [a1, . . . am] , b3, . . . bn]
= νm [μn [b1, a1, b3, . . . bn] , μn [b1, a2, b3, . . . bn] , . . . μn [b1, am, b3, . . . bn]]
(16.2)
...
μn [b1, b2, . . . bn−1, νm [a1, . . . am]]
= νm [μn [b1, b2, . . . bn−1, a1] , μn [b1, b2, . . . bn−1, a2] , . . . μn [b1, b2, . . . bn−1, am]] ,
(16.3)
where ai, bj ∈ R.
99

It is seen that the operations μn and νm enter above in a non-symmetric way,
which allows us to distinguish them: one of them (μn, the n-ary multiplication)
“distributes” over the other one νm, and therefore νm is called the addition.
If only some of the relations (16.1)-(16.3) hold, then such distributivity is partial
(the analog of left and right distributivity in the binary case).
Remark 16.2. The operations μn and νm need have nothing to do with ordinary
multiplication (in the binary case denoted by μ2 =⇒ (∙)) and addition (in the
binary case denoted by ν2 =⇒ (+)).
Example 16.3. Let A = R, n = 2, m = 3, and μ2 [b1, b2] = bb2
1 ,
ν3 [a1, a2, a3] = a1a2a3 (ordinary product in R). The partial distributivity now
is (a1a2a3)
b2
= ab2
1 ab2
2 ab2
3 (only the first relation (16.1) holds).
100

Let both operations μn and νm be (totally) associative, which (in our definition
DUPLIJ [2012]) means independence of the composition of two operations under
placement of the internal operations (there are n and m such placements and
therefore (n + m) corresponding relations)
μn [a, μn [b] , c] = invariant, (16.4)
νm [d, νm [e] , f] = invariant, (16.5)
where the polyads a, b, c, d, e, f have corresponding length, and then both
R | μn | assoc and R | νm | assoc are polyadic semigroups Sn and Sm.
A commutative semigroup A | νm | assoc, comm is defined by
νm [a] = νm [σ ◦ a], for all σ ∈ Sn, where Sn is the symmetry group.
If the equation νm [a, x, b] = c is solvable for any place of x, then
R | νm | assoc, solv is a polyadic group Gm.
101

Definition 16.4. A polyadic (m, n)-ring Rm,n is a set R with two operations
μn : Rn
→ R and νm : Rm
→ R, such that:
1) they are distributive (16.1)-(16.3);
2) R | μn | assoc is a n-ary semigroup;
3) R | νm | assoc, comm, solv is a commutative m-ary group.
It is obvious that a (2, 2)-ring R2,2 is an ordinary (binary) ring.
Polyadic rings have much richer structure and unusual properties CELAKOSKI
[1977], CROMBEZ [1972], ˇCUPONA [1965], LEESON AND BUTSON [1980].
If the multiplicative semigroup R | μn | assoc is commutative,
μn [a] = μn [σ ◦ a], for all σ ∈ Sn, then Rm,n is called a commutative
polyadic ring.
If it contains the identity, then Rm,n is a (m, n)-semiring.
If the distributivity is only partial, then Rm,n is called a polyadic near-ring.
102

A polyadic ring is derived, if νm and μn are equivalent to a repetition of the
binary addition and multiplication, while R | + and R | ∙ are commutative
(binary) group and semigroup.
An n-admissible “length of word (x)” should be congruent to 1 mod (n − 1),
containing μ (n − 1) + 1 elements ( μ is a “number of multiplications”)
μ
( μ)
n [x] (x ∈ R μ(n−1)+1
), or polyads. An m-admissible “quantity of words
(y)” in a polyadic “sum” has to be congruent to 1 mod (m − 1), i.e. consisting
of ν (m − 1) + 1 summands ( ν is a “number of additions”) ν
( ν )
m [y]
(y ∈ R ν (m−1)+1
).
“Polyadization” of a binary expression (m = n = 2): the multipliers
μ + 1 → μ (n − 1) + 1 and summands ν + 1 → ν (m − 1) + 1.
Example 16.5. “Trivial polyadization”: the simplest (m, n)-ring derived from the
ring of integers Z as the set of ν (m − 1) + 1 “sums” of n-admissible
( μ (n − 1) + 1)-ads (x), x ∈ Z μ(n−1)+1
LEESON AND BUTSON [1980].
103

The additive m-ary polyadic power and multiplicative n-ary polyadic power are
(inside polyadic products we denote repeated entries by
k
x, . . . , x as xk
)
x ν +m = ν( ν )
m x ν (m−1)+1
, x μ ×n = μ( μ)
n x μ(n−1)+1
, x ∈ R,
(16.6)
Polyadic and ordinary powers differ by 1: x ν +2 = x ν +1
, x μ ×2 = x μ+1
.
The polyadic idempotents in Rm,n satisfy
x ν +m = x, x μ ×n = x, (16.7)
and are called the additive ν-idempotent and the multiplicative μ-idempotent.
The idempotent zero z ∈ R, is (if it exists) defined by
νm [x, z] = z, ∀x ∈ Rm−1
. (16.8)
If a zero z exists, it is unique.
An element x is nilpotent, if
x 1 +m = z. (16.9)
104

The unit e of Rm,n is multiplicative 1-idempotent
μn en−1
, x = x, ∀x ∈ R. (16.10)
In case of a noncommutative polyadic ring x can be on any place.
In distinction with the binary case there are unusual polyadic rings :
1) with no unit and no zero (zeroless, nonunital);
2) with several units and no zero;
3) with all elements are units.
In polyadic rings invertibility is not connected with unit and zero elements.
For a fixed element x ∈ R its additive querelement ˜x and multiplicative
querelement ˉx are defined by
νm xm−1
, ˜x = x, μn xn−1
, ˉx = x, (16.11)
Because R | νm is a commutative group, each x ∈ R has its additive
querelement ˜x (and is querable or “polyadically invertible”).
105

The n-ary semigroup R | μn can have no multiplicatively querable elements.
If each x ∈ R has its unique querelement, then R | μn is an n-ary group.
Denote R∗
= R {z}, if the zero z exists.
If R∗
| μn is the n-ary group, then Rm,n is a (m, n)-division ring.
Definition 16.6. A commutative (m, n)-division ring is a (m, n)-field Fm,n.
Example 16.7. a) The set iR with i2
= −1 is a (2, 3)-field with no unit and a
zero 0 (operations in C), the multiplicative querelement of ix is −i x (x = 0).
b) The set of fractions ix/y | x, y ∈ Zodd
, i2
= −1 is a (3, 3)-field with no
zero and no unit (operations are in C), while the additive and multiplicative
querelements of ix/y are −ix/y and −iy/x, respectively.
c) The set of antidiagonal 2 × 2 matrices over R is a (2, 3)-field with zero
z =


0 0
0 0

, two units e = ±


0 1
1 0

, but the multiplicative querelement of


0 x
y 0

 is


0 1/y
1/x 0

.
106

17 Polyadic analogs of integer number ring Z and field Z/pZ
The theory of finite fields LIDL AND NIEDERREITER [1997] plays a very important
role. Peculiarities:
1) Uniqueness - they can have only special numbers of elements (the order is any
power of a prime integer pr
) and this fully determines them, all finite fields of the
same order are isomorphic;
2) Existence of their “minimal” (prime) finite subfield of order p, which is
isomorphic to the congruence class of integers Z pZ.
We propose a special version of the (prime) finite fields: instead of the binary ring
of integers Z, we consider a polyadic ring.
The concept of the polyadic integer numbers Z(m,n) as representatives of a fixed
congruence class, forming the (m, n)-ring (with m-ary addition and n-ary
multiplication), was introduced in DUPLIJ [2017a].
107

We define new secondary congruence classes and the corresponding finite
(m, n)-rings Z(m,n) (q) of polyadic integer numbers, which give Z qZ in the
“binary limit”. We construct the prime polyadic fields F(m,n) (q), which can be
treated as polyadic analog of the Galois field GF (p).
Ring of polyadic integer numbers Z(m,n)
The ring of polyadic integer numbers Z(m,n) was introduced in DUPLIJ [2017a].
Consider a congruence class (residue class) of an integer a modulo b
[[a]]b = {{a + bk} | k ∈ Z, a ∈ Z+, b ∈ N, 0 ≤ a ≤ b − 1} . (17.1)
Denote a representative by xk = x
[a,b]
k = a + bk, where {xk} is an infinite set.
108

Informally, there are two ways to equip (17.1) with operations:
1. The “External” way: to define operations between the classes [[a]]b. Denote
the class representative by [[a]]b ≡ a , and introduce the binary operations
+ , ∙ as
a1 + a2 = (a1 + a2) , (17.2)
a1 ∙ a2 = (a1a2) . (17.3)
The binary residue class ring is defined by
Z bZ = {{a } | + , ∙ , 0 , 1 } . (17.4)
With prime b = p, the ring Z pZ is a binary finite field having p elements.
This is the standard finite field theory LIDL AND NIEDERREITER [1997].
109

2. The “Internal” way is to introduce (polyadic) operations inside a class [[a]]b
(with both a and b fixed). Define the commutative m-ary addition and
commutative n-ary multiplication of representatives xki
in [[a]]b by
νm [xk1
, xk2
, . . . , xkm
] = xk1
+ xk2
+ . . . + xkm
, (17.5)
μn [xk1 , xk2 , . . . , xkn ] = xk1 xk2 . . . xkn , xki ∈ [[a]]b , ki ∈ Z.
(17.6)
Remark 17.1. Binary sums xk1
+ xk2 and products xk1
xk2 are not in
[[a]]b for arbitrary a ∈ Z+, b ∈ N, 0 ≤ a ≤ b − 1.
110

Proposition 17.2 ( DUPLIJ [2017b]). The polyadic operations νm and μn
closed in [[a]]b, if the arities (m, n) have the minimal values satisfying
ma ≡ a (mod b) , (17.7)
an
≡ a (mod b) . (17.8)
Remark 17.3. If n = b = p is prime, then (17.8) is valid for any a ∈ N,
which is another formulation of Fermat’s little theorem.
Definition 17.4 ( DUPLIJ [2017a]). The congruence class [[a]]b equipped
with a structure of nonderived infinite commutative polyadic ring is called a
(m, n)-ring of polyadic integer numbers
Z(m,n) ≡ Z
[a,b]
(m,n) = {[[a]]b | νm, μn} . (17.9)
Definition 17.5. A polyadic prime number is such that obeys a unique expansion
xkp
= μ( )
n xkp
, e (n−1)
, (17.10)
where e a polyadic unit of Z(m,n) (if exists).
111

Example 17.6. In the residue class
[[3]]4 = {. . . − 25, −21, −17, −13, −9, −5, −1, 3, 7, 11, 15, 19, 23, 27, 31 . . .}
(17.11)
To retain the same class [[3]]4, we can add 4 ν + 1 = 5, 9, 13, 17, . . .
representatives and multiply 2 μ + 1 = 3, 5, 7, 9, . . . representatives only.
E.g., take “number” of additions ν = 2, and multiplications μ = 3, to get
(7 + 11 + 15 + 19 + 23) − 5 − 9 − 13 − 17 = 31 = 3 + 4 ∙ 7 ∈ [[3]]4,
((7 ∙ 3 ∙ 11) ∙ 19 ∙ 15) ∙ 31 ∙ 27 = 55 103 895 = 3 + 4 ∙ 13775973 ∈ [[3]]4.
This means that [[3]]4 is the polyadic (5, 3)-ring Z(5,3) = Z
[3,4]
(5,3).
Remark 17.7. Elements of the (m, n)-ring Z(m,n)(polyadic integer numbers)
are not ordinary integers (forming a (2, 2)-ring). A representative x
[a,b]
k , e.g.
3 = 3(5,3) ∈ Z
[3,4]
(5,3) is different from 3 = 3(3,2) ∈ Z
[1,2]
(3,2), and different from the
binary 3 ∈ Z ≡ Z
[0,1]
(2,2), i.e. properly speaking 3(5,3) = 3(3,2) = 3, since their
operations (multiplication and addition) are different.
112

The parameters-to-arity mapping
Remark 17.8. a) Solutions to (17.7) and (17.8) do not exist for all a and b;
b) The pair a, b determines m, n uniquely;
c) For several different pairs a, b there can be the same arities m, n.
Assertion 17.9. The parameters-to-arity mapping ψ : (a, b) −→ (m, n) is a
partial surjection.
The characterization of the fixed congruence class [[a]]b and the corresponding
(m, n)-ring of polyadic integer numbers Z
[a,b]
(m,n) can be done in terms of the
shape invariants I, J ∈ Z+ defined uniquely by (TABLE 3 in DUPLIJ [2017a])
I = I[a,b]
m = (m − 1)
a
b
, J = J[a,b]
n =
an
− a
b
. (17.12)
In the binary case, when m = n = 2 (a = 0, b = 1), both shape invariants
vanish, I = J = 0. There exist “partially” binary cases, when only n = 2 and
m = 2, while J is nonzero, for instance in Z
[6,10]
(6,2) we have I = J = 3.
113

Polyadic rings of secondary classes
A special method of constructing a finite nonderived polyadic ring by combining
the “External” and “Internal” methods was given in DUPLIJ [2017b].
Introduce the finite polyadic ring Z(m,n) cZ, where Z(m,n) is a polyadic ring.
If we directly consider the “double” class {a + bk + cl} and fix a and b, then the
factorization by cZ will not give a closed operations for arbitrary c.
Assertion 17.10. If the finite polyadic ring Z
[a,b]
(m,n) cZ has q elements, then
c = bq. (17.13)
Definition 17.11. A secondary (equivalence) class of a polyadic integer
x
[a,b]
k = a + bk ∈ Z
[a,b]
(m,n) “modulo” bq (with q being the number of
representatives x
[a,b]
k , for fixed b ∈ N and 0 ≤ a ≤ b − 1) is
x
[a,b]
k
bq
= {{(a + bk) + bql} | l ∈ Z, q ∈ N, 0 ≤ k ≤ q − 1} .
(17.14)
114

Remark 17.12. In the binary limit a = 0, b = 1 and Z
[0,1]
(2,2) = Z, the secondary
class becomes the ordinary class (17.1).
If the values of a, b, q are clear from the context, we denote the secondary class
representatives by an integer with two primes x
[a,b]
k
bq
≡ xk ≡ x .
Example 17.13. a) For a = 3, b = 6 and for 4 elements and k = 0, 1, 2, 3
x
[3,6]
k
24
= 3 , 9 , 15 , 21 , ([[k]]4 = 0 , 1 , 2 , 3 ) . (17.15)
b) If a = 4, b = 5, for 3 elements and k = 0, 1, 2 we get
x
[4,5]
k
15
= 4 , 9 , 14 , ([[k]]3 = 0 , 1 , 2 ) . (17.16)
115

c) Let a = 3, b = 5, then for q = 4 elements we have the secondary classes
with k = 0, 1, 2, 3 (the binary limits are in brackets)
x
[3,5]
k
20
= 3 , 8 , 13 , 18 =



3 = {. . . − 17, 3, 23, 43, 63, . . .} ,
8 = {. . . − 12, 8, 28, 48, 68, . . .} ,
13 = {. . . − 7, 13, 33, 53, 73, . . .} ,
18 = {. . . − 2, 18, 38, 58, 78, . . .} ,
(17.17)







[[k]]4 = 0 , 1 , 2 , 3 =



0 = {. . . − 4, 0, 4, 8, 12, . . .} ,
1 = {. . . − 3, 1, 5, 9, 13, . . .} ,
2 = {. . . − 2, 2, 6, 10, 14, . . .} ,
3 = {. . . − 1, 3, 7, 11, 15, . . .} .







(17.18)
Difference between classes: 1) they are described by rings of different arities; 2)
some of them are fields.
116

Finite polyadic rings
Now we determine the nonderived polyadic operations between secondary
classes which lead to finite polyadic rings.
Proposition 17.14. The set {xk} of q secondary classes k = 0, . . . , q − 1
(with the fixed a, b) can be endowed with the commutative m-ary addition
xkadd
= νm xk1
, xk1
, . . . , xkm
, (17.19)
kadd ≡ (k1 + k2 + . . . + km) + I[a,b]
m (mod q) (17.20)
and commutative n-ary multiplication
xkmult
= μn xk1
, xk1
, . . . , xkn
, (17.21)
kmult ≡ an−1
(k1 + k2 + . . . + kn) + an−2
b (k1k2 + k2k3 + . . . + kn−1kn) + . . .
+bn−1
k1 . . . kn + J[a,b]
n (mod q) , (17.22)
satisfying the polyadic distributivity, shape invariants I
[a,b]
m , J
[a,b]
n are in (17.12).
117

Definition 17.15. The set of secondary classes (17.14) equipped with operations
(17.19), (17.21) is denoted by
Z(m,n) (q) ≡ Z
[a,b]
(m,n) (q) = Z
[a,b]
(m,n) (bq) Z = {{xk} | νm, μn} , (17.23)
and is a finite secondary class (m, n)-ring of polyadic integer numbers
Z(m,n) ≡ Z
[a,b]
(m,n). The value q (the number of elements) is called its order.
Example 17.16. a) In (5, 3)-ring Z
[3,4]
(4,3) (2) with 2 secondary classes all
elements are units (marked by subscript e) e1 = 3e = 3 , e2 = 7e = 7 ,
because
μ3 [3 , 3 , 3 ] = 3 , μ3 [3 , 3 , 7 ] = 7 , μ3 [3 , 7 , 7 ] = 3 , μ3 [7 , 7 , 7 ] = 7 .
(17.24)
b) The ring Z
[5,6]
(7,3) (4) consists of 4 units
e1 = 5e , e2 = 11e , e3 = 17e , e4 = 23e , and no zero.
Remark 17.17. Equal arity finite polyadic rings of the same order Z
[a1,b1]
(m,n) (q)
and Z
[a2,b2]
(m,n) (q) may be not isomorphic.
118

Example 17.18. The finite polyadic ring Z
[1,3]
(4,2) (2) of order 2 consists of unit
e = 1e = 1 and zero z = 4z = 4 only,
μ2 [1 , 1 ] = 1 , μ2 [1 , 4 ] = 4 , μ2 [4 , 4 ] = 4 , (17.25)
and therefore Z
[1,3]
(4,2) (2) is a field, because {1 , 4z } 4z is a (trivial) binary
group, consisting of one element 1e .
However, Z
[4,6]
(4,2) (2) has the zero z = 4z = 4 , 10 and has no unit, because
μ2 [4 , 4 ] = 4 , μ2 [4 , 10 ] = 4 , μ2 [10 , 10 ] = 4 , (17.26)
so that Z
[4,6]
(4,2) (2) is not a field, because the nonzero element 10 is nilpotent.
Their additive 4-ary groups are also not isomorphic, while Z
[1,3]
(4,2) (2) and
Z
[4,6]
(4,2) (2) have the same arity (m, n) = (4, 2) and order 2.
Assertion 17.19. For a fixed arity shape (m, n), there can be non-isomorphic
secondary class polyadic rings Z(m,n) (q) of the same order q, which describe
different binary residue classes [[a]]b.
119

Finite polyadic fields
Proposition 17.20. A finite polyadic ring Z
[a,b]
(m,n) (q) is a secondary class finite
(m, n)-field F
[a,b]
(m,n) (q) if all its elements except z (if it exists) are polyadically
multiplicative invertible having a unique querelement.
In the binary case LIDL AND NIEDERREITER [1997] the residue (congruence)
class ring (17.4) with q elements Z qZ is a congruence class (non-extended)
field, if its order q = p is a prime number, such that
F (p) = [[a]]p | + , ∙ , 0 , 1 , a = 0, 1, . . . , p − 1.
All non-extended binary fields of a fixed prime order p are isomorphic, and so it is
natural to study them in a more “abstract” way. The mapping Φp [[a]]p = a is
an isomorphism of binary fields Φp : F (p) → F (p), where
F (p) = {{a} | +, ∙, 0, 1}mod p is an “abstract” non-extended (prime) finite field
of order p (or Galois field GF (p)).
120

Consider the set of polyadic integer numbers
{xk} ≡ x
[a,b]
k = {a + bk} ∈ Z
[a,b]
(m,n), b ∈ N and
0 ≤ a ≤ b − 1, 0 ≤ k ≤ q − 1, q ∈ N, which obey the operations
(17.5)–(17.6).
Definition 17.21. The “abstract” non-extended finite (m, n)-field of order q is
F(m,n) (q) ≡ F
[a,b]
(m,n) (q) = {{a + bk} | νm, μn}mod bq , (17.27)
if {{xk} | νm}mod bq is an additive m-ary group, and {{xk} | μn}mod bq (or,
when zero z exists, {{xk z} | μn}mod bq) is a multiplicative n-ary group.
Define a one-to-one onto mapping from the secondary congruence class to its
representative by Φ
[a,b]
q x
[a,b]
k
bq
= x
[a,b]
k and arrive
Proposition 17.22. The mapping Φ
[a,b]
q : F
[a,b]
(m,n) (q) → F
[a,b]
(m,n) (q) is a
polyadic ring homomorphism (being, in fact, an isomorphism).
In TABLE 2 we present the “abstract” non-extended polyadic finite fields
F
[a,b]
(m,n) (q) of lowest arity shape (m, n) and orders q.
121

Table 2: The finite polyadic rings Z
[a,b]
(m,n) (q) and (m, n)-fields F
[a,b]
m,n (q).
a b 2 3 4 5 6
1
m = 3
n = 2
1e,3
1e,3z,5
1e,3,5,7
q=5,7,8
m = 4
n = 2
1e,4z
1e,4,7
1e,4z,7,10
q=5,7,9
m = 5
n = 2
1e,5
1e,5,9z
1e,5,9,13
q=5,7,8
m = 6
n = 2
1e,6z
1e,6z,11
1e,6,11,16z
q=5,7
m = 7
n = 2
1e,7
1e,7,13
1e,7,13,19
q=5,6,7,8,9
2
m = 4
n = 3
2z,5e
2,5,8e
2,5e,8z,11e
q=5,7,9
m = 6
n = 5
2z,7e
2e,7,12z
2,7e,12z,17e
q=5,7
m = 4
n = 3
2,8z
2,8e,14
2,8z,14,20
q=5,7,9
3
m = 5
n = 3
3e,7e
3z,7e,11e
3,7e,11,15e
q=5,6,7,8
m = 6
n = 5
3e,8z
3z,8e,13e
3e,8z,13e,18
q=5,7
m = 3
n = 2
3,9e
3,9z,15
3,9e,15,21
q=5,7,8
122

In the multiplicative structure the following crucial differences between the binary
finite fields F (q) and polyadic fields F(m,n) (q) can be outlined.
1. The order of a non-extended finite polyadic field may not be prime (e.g.,
F
[1,2]
(3,2) (4), F
[3,4]
(5,3) (8), F
[2,6]
(4,3) (9)), and may not even be a power of a prime
binary number (e.g. F
[5,6]
(7,3) (6), F
[3,10]
(11,5) (10)), and see TABLE 3.
2. The polyadic characteristic χp of a non-extended finite polyadic field can
have values such that χp + 1 (corresponding in the binary case to the
ordinary characteristic χ) can be nonprime.
3. There exist finite polyadic fields with more than one unit, and also all
elements can be units. Such cases are marked in TABLE 3 by subscripts
which indicate the number of units.
4. The(m, n)-fields can be zeroless-nonunital, but have unique additive and
multiplicative querelements: [[a]]b | νm , [[a]]b | μn are polyadic groups.
5. The zeroless-nonunital polyadic fields are totally (additively and
multiplicatively) nonderived.
123

Example 17.23. 1) The zeroless-nonunital polyadic finite fields having lowest
|a + b| are, e.g., F
[3,8]
(9,3) (2), F
[3,8]
(9,3) (4), F
[5,8]
(9,3) (4), F
[5,8]
(9,3) (8), also F
[4,9]
(10,4) (3),
F
[4,9]
(10,4) (9), and F
[7,9]
(10,4) (3), F
[7,9]
(10,4) (9).
2) The multiplication of the zeroless-nonunital (9, 3)-field F
[5,8]
(9,3) (2) is
μ3 [5, 5, 5] = 13, μ3 [5, 5, 13] = 5, μ3 [5, 13, 13] = 13, μ3 [13, 13, 13] = 5.
The (unique) multiplicative querelements ˉ5 = 13, 13 = 5. The addition is
ν9 5
9
= 13, ν9 5
8
, 13 = 5, ν9 5
7
, 13
2
= 13, ν9 5
6
, 13
3
= 5, ν9 5
5
, 13
4
= 13,
ν9 5
4
, 13
5
= 5, ν9 5
3
, 13
6
= 13, ν9 5
2
, 13
7
= 5, ν9 5, 13
8
= 13, ν9 13
9
= 5.
The additive (unique) querelements are ˜5 = 13, 13 = 5. So all elements are
additively and multiplicatively querable (polyadically invertible), and therefore ν9
is 9-ary additive group operation and μ3 is 3-ary multiplicative group operation,
as it should be for a field. Because it contains no unit and no zero, F
[5,8]
(9,3) (2) is
actually a zeroless-nonunital finite (9, 3)-field of order 2.
124

Example 17.24. The (4, 3)-ring Z
[2,3]
(4,3) (6) is zeroless, and [[3]]4 | ν4 is its
4-ary additive group (each element has a unique additive querelement). Despite
each element of [[2]]3 | μ3 having a querelement, it is not a multiplicative 3-ary
group, because for the two elements 2 and 14 we have nonunique querelements
μ3 [2, 2, 5] = 2, μ3 [2, 2, 14] = 2, μ3 [14, 14, 2] = 14, μ3 [14, 14, 11] = 14.
(17.28)
Example 17.25. The polyadic (9, 3)-fields corresponding to the congruence
classes [[5]]8 and [[7]]8 are not isomorphic for orders q = 2, 4, 8 (see TABLE 3).
Despite both being zeroless, the first F
[5,8]
(9,3) (q) are nonunital, while the second
F
[7,8]
(9,3) (q) has two units, which makes an isomorphism impossible.
125

Polyadic field order
In binary case the order of an element x ∈ F (p) is defined as a smallest integer
λ such that xλ
= 1. Obviously, the set of fixed order elements forms a cyclic
subgroup of the multiplicative binary group of F (p), and λ | (p − 1).
If λ = p − 1, such an element is called a primitive (root), it generates all
elements, and these exist in any finite binary field.
Any element of F (p) is idempotent xp
= x, while all its nonzero elements satisfy
xp−1
= 1 (Fermat’s little theorem).
A non-extended (prime) finite field is fully determined by its order p (up to
isomorphism), and, moreover, any F (p) is isomorphic to Z pZ.
In the polyadic case, the situation is more complicated. Because the related
secondary class structure (17.27) contains parameters in addition to the number
of elements q, the order of (non-extended) polyadic fields may not be prime, or
nor even a power of a prime integer (e.g. F
[5,6]
(7,3) (6) or F
[3,10]
(11,5) (10)).
126

Because finite polyadic fields can be zeroless, nonunital and have many (or even
all) units (see TABLE 3), we cannot use units in the definition of the element order.
Definition 17.26. If x ∈ F(m,n) (q) satisfies
x λp ×n = x, (17.29)
then the smallest such λp is called the idempotence polyadic order ord x = λp.
Definition 17.27. The idempotence polyadic order λ[a,b]
p of a finite polyadic field
F
[a,b]
(m,n) (q) is the maximum λp of all its elements, we call such field
λ[a,b]
p -idempotent and denote ord F
[a,b]
(m,n) (q) = λ[a,b]
p .
In TABLE 3 we present the idempotence polyadic order λ[a,b]
p for small a, b.
Definition 17.28. Denote by q∗ the number of nonzero distinct elements in
F(m,n) (q)
q∗ =



q − 1, if ∃z ∈ F(m,n) (q)
q, if z ∈ F(m,n) (q) ,
(17.30)
which is called a reduced (field) order (in binary case we have the first line only).
127

Table 3: Idempotence polyadic orders λ[a,b]
p for finite polyadic fields F
[a,b]
(m,n) (q).
[[a]]b Arities Finite polyadic field order q
b a (m, n) 2 3 4 5 6 7 8 9 10
2 1 (3, 2) 2 2 2 4 ∅ 6 4 ∅ ∅
3 1 (4, 2) 1 3 ∅ 4 ∅ 6 ∅ 9 ∅
2 (4, 3) 1 3 ∅ 22e ∅ 32e ∅ 9 ∅
4 1 (5, 2) 2 2 4 4 ∅ 6 8 ∅ ∅
3 (5, 3) 12e 12e 22e 22e ∅ 32e 42e ∅ ∅
5 1 (6, 2) 1 2 ∅ 5 ∅ 6 ∅ ∅ ∅
2 (6, 5) 1 12e ∅ 5 ∅ 32e ∅ ∅ ∅
3 (6, 5) 1 12e ∅ 5 ∅ 32e ∅ ∅ ∅
4 (6, 3) 1 12e ∅ 5 ∅ 32e ∅ ∅ ∅
6 1 (7, 2) 2 3 2 4 6 6 4 9 ∅
2 (4, 3) ∅ 3 ∅ 22e ∅ 32e ∅ 9 ∅
3 (3, 2) 2 ∅ 2 4 ∅ 6 4 ∅ ∅
4 (4, 2) ∅ 3 ∅ 4 ∅ 6 ∅ 9 ∅
5 (7, 3) 12e 3 14e 22e 32e 32e 2 9 ∅
7 1 (8, 2) 1 2 ∅ 4 ∅ 7 ∅ ∅ ∅
2 (8, 4) 1 2 ∅ 4 ∅ 7 ∅ ∅ ∅
3 (8, 7) 1 12e ∅ 22e ∅ 7 ∅ ∅ ∅
4 (8, 4) 1 2 ∅ 4 ∅ 7 ∅ ∅ ∅
5 (8, 7) 1 12e ∅ 22e ∅ 7 ∅ ∅ ∅
6 (8, 3) 1 12e ∅ 22e ∅ 7 ∅ ∅ ∅
8 1 (9, 2) 2 2 4 4 ∅ 6 8 ∅ ∅
3 (9, 3) 2 12e 4 22e ∅ 32e 8 ∅ ∅128

Theorem 17.29. If a finite polyadic field F(m,n) (q) has an order q, such that
q∗ = qadm
∗ = (n − 1) + 1 is n-admissible, then (for n ≥ 3 and one unit):
1. A sequence of the length q∗ (n − 1) built from any fixed element
y ∈ F(m,n) (q) is neutral
μ(q∗)
n x, yq∗(n−1)
= x, ∀x ∈ F(m,n) (q) . (17.31)
2. Any element y satisfies the polyadic idempotency condition
y q∗ ×n = y, ∀y ∈ F(m,n) (q) . (17.32)
Finite polyadic fields F
[a,b]
(m,n) (q) having n-admissible reduced order
q∗ = qadm
∗ = (n − 1) + 1 ( ∈ N) (underlined in TABLE 3) are closest to the
binary finite fields F (p) in their general properties: they are always half-derived,
while if they additionally contain a zero, they are fully derived.
If q∗ = qadm
∗ , then F
[a,b]
(m,n) (q) can be nonunital or contain more than one unit
(subscripts in TABLE 3).
129

Assertion 17.30. The finite fields F
[a,b]
(m,n) (q) of n-admissible reduced order
q∗ = qadm
∗ cannot have more than one unit and cannot be zeroless-nonunital.
Assertion 17.31. If q∗ = qadm
∗ , and F
[a,b]
(m,n) (q) is unital zeroless, then the
reduced order q∗ is the product of the idempotence polyadic field order
λ[a,b]
p = ord F
[a,b]
(m,n) (q) and the number of units κe (if a b and n ≥ 3)
q∗ = λ[a,b]
p κe. (17.33)
Structure of the multiplicative group G
[a,b]
n (q∗) of F
[a,b]
(m,n) (q)
Some properties of commutative cyclic n-ary groups were considered for
particular relations between orders and arity. Here we have: 1) more parameters
and different relations between these, the arity m, n and order q; 2) the
(m, n)-field under consideration, which leads to additional restrictions. In such a
way exotic polyadic groups and fields arise which have unusual properties that
have not been studied before.
130

Definition 17.32. An element xprim ∈ G
[a,b]
n (q∗) is called n-ary primitive, if its
idempotence order is
λp = ord xprim = q∗. (17.34)
All λp polyadic powers x
1 ×n
prim , x
2 ×n
prim , . . . , x
q∗ ×n
prim ≡ xprim generate other
elements, and so G
[a,b]
n (q∗) is a finite cyclic n-ary group generated by xprim,
i.e. G
[a,b]
n (q∗) = x
i ×n
prim | μn . Number primitive elements in κprim.
Assertion 17.33. For zeroless F
[a,b]
(m,n) (q) and prime order q = p, we have
λ[a,b]
p = q, and G
[a,b]
n (q) is indecomposable (n ≥ 3).
Example 17.34. The smallest 3-admissible zeroless polyadic field is F
[2,3]
(4,3) (3)
with the unit e = 8e and two 3-ary primitive elements 2, 5 having 3-idempotence
order ord 2 = ord 5 = 3, so κprim = 2 , because
2 1 ×3 = 8e, 2 2 ×3 = 5, 2 3 ×3 = 2, 5 1 ×3 = 8e, 5 2 ×3 = 2, 5 3 ×3 = 5,
(17.35)
and therefore G
[2,3]
3 (3) is a cyclic indecomposable 3-ary group.
131

Assertion 17.35. If F
[a,b]
(m,n) (q) is zeroless-nonunital, then every element is
n-ary primitive, κprim = q, also λ[a,b]
p = q (the order q can be not prime), and
G
[a,b]
n (q) is a indecomposable commutative cyclic n-ary group without identity
(n ≥ 3).
Example 17.36. The (10, 7)-field F
[5,9]
(10,7) (9) is zeroless-nonunital, each
element (has λp = 9) is primitive and generates the whole field, and therefore
κprim = 9, thus the 7-ary multiplicative group G
[5,9]
7 (9) is indecomposable and
without identity.
The structure of G
[a,b]
n (q∗) can be extremely nontrivial and may have no analogs
in the binary case.
Assertion 17.37. If there exists more than one unit, then:
1. If G
[a,b]
n (q∗) can be decomposed on its n-ary subroups, the number of units
κe coincides with the number of its cyclic n-ary subgroups
G
[a,b]
n (q∗) = G1 ∪ G2 . . . ∪ Gke
which do not intersect Gi ∩ Gj = ∅,
i, j = i = 1, . . . , κe, i = j.
132

2. If a zero exists, then each Gi has its own unit ei, i = 1, . . . , κe.
3. In the zeroless case G
[a,b]
n (q) = G1 ∪ G2 . . . ∪ Gke ∪ E (G), where
E (G) = {ei} is the split-off subgroup of units.
Example 17.38. 1) In the (9, 3)-field F
[5,8]
(9,3) (7) there is a single zero z = 21z
and two units e1 = 13e, e2 = 29e, and so its multiplicative 3-ary group
G
[5,8]
3 (6) = {5, 13e, 29e, 37, 45, 53} consists of two nonintersecting (which is
not possible in the binary case) 3-ary cyclic subgroups G1 = {5, 13e, 45} and
G2 = {29e, 37, 53} (for both λp = 3)
G1 = 5 1 ×3 = 13e, 5 2 ×3 = 45, 5 3 ×3 = 5 , ˉ5 = 45, 45 = 5,
G2 = 37 1 ×3 = 29e, 37 2 ×3 = 53, 37 3 ×3 = 37 , 37 = 53, 53 = 37.
All nonunital elements in G
[5,8]
3 (6) are (polyadic) 1-reflections, because
5 1 ×3 = 45 1 ×3 = 13e and 37 1 ×3 = 53 1 ×3 = 29e, and so the subgroup
of units E (G) = {13e, 29e} is unsplit E (G) ∩ G1,2 = ∅.
133

2) For the zeroless F
[7,8]
(9,3) (8), its multiplicative 3-group
G
[5,8]
3 (6) = {7, 15, 23, 31e, 39, 47, 55, 63e} has two units e1 = 31e,
e2 = 63e, and it splits into two nonintersecting nonunital cyclic 3-subgroups
(λp = 4 and λp = 2) and the subgroup of units
G1 = 7 1 ×3 = 23, 7 2 ×3 = 39, 7 3 ×3 = 55, 7 4 ×3 = 4 ,
ˉ7 = 55, 55 = 7, 23 = 39, 39 = 23,
G2 = 15 1 ×3 = 47, 15 2 ×3 = 15 , 15 = 47, 47 = 15,
E (G) = {31e, 63e} .
There are no μ-reflections, and so E (G) splits out E (G) ∩ G1,2 = ∅.
If all elements are units E (G) = G
[a,b]
n (q), the group is 1-idempotent λp = 1.
Assertion 17.39. If F
[a,b]
(m,n) (q) is zeroless-nonunital, then there no n-ary cyclic
subgroups in G
[a,b]
n (q).
134

The subfield structure of F
[a,b]
(m,n) (q) can coincide with the corresponding
subgroup structure of the multiplicative n-ary group G
[a,b]
n (q∗), only if its additive
m-ary group has the same subgroup structure. However, we have
Assertion 17.40. Additive m-ary groups of all polyadic fields F
[a,b]
(m,n) (q) have
the same structure: they are polyadically cyclic and have no proper m-ary
subgroups.
Therefore, in additive m-ary groups each element generates all other elements,
i.e. it is a primitive root.
Theorem 17.41. The polyadic field F
[a,b]
(m,n) (q), being isomorphic to the
(m, n)-field of polyadic integer numbers Z
[a,b]
(m,n) (q), has no any proper subfield.
In this sense, F
[a,b]
(m,n) (q) can be named a prime polyadic field.
135

S. Duplij. Polyadic algebraic structures and their applications

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