Polyadic systems and their multiplace representations

Polyadic Systems and their

Multiplace Representations

STEVEN DUPLIJ

Institute of Mathematics, University of Munster

http://wwwmath.uni-muenster.de/u/duplij

2014

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 DORNTE

[1929] (inspired by E. Nother).

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-Hosszu

theorem HOSSZU [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 wereconsidered, by the SD in BOROWIEC ET AL. [2006], DUPLIJ [2013a].

Some theorems connecting representations of binary and n-arygroups were presented in DUDEK AND SHAHRYARI [2012].

In physics, the most applicable structures are the nonassociativeGrassmann, Clifford and Lie algebras LOHMUS ET AL. [1994],GEORGI [1999]. The ternary analog of Clifford algebra wasconsidered in ABRAMOV [1995], and the ternary analog ofGrassmann algebra ABRAMOV [1996] was exploited to constructternary extensions of supersymmetry ABRAMOV ET AL. [1997].

In the higher arity studies, the standard Lie bracket is replaced by alinear n-ary bracket, and the algebraic structure of thecorresponding model is defined by the additional characteristicidentity for this generalized bracket, corresponding to the Jacobiidentity 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 Hosszu-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.

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) 7−→ gi.

The i-diagonal Diagn : G→ G×n sends one element to the equal

element n-tuple g 7−→ (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 ε.

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)

7

3 Notations

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)

because g is a polyad of length (n− 1). Visually, it is seen from the

commutative diagram

G×(n−1) × {•}id×(n−1) ×μ(c)0 G×n

G×(n−1)

ε

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 Preliminaries

4 Preliminaries

Definition 4.1. A polyadic system G 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 | μ′n′〉 over the same set G, which has

multiplication with a different arity n′.

9

4 Preliminaries

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 n′iter . Denote the number of

iterating multiplications by `μ, and use the bold Greek letters

μ`μn for the resulting composition of n-ary multiplications

μ′n′ = μ`μn

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`μn [g]. The operation μ`μn is named

a long product DORNTE [1929] or derived DUDEK [2007].

10

4 Preliminaries

2. Reducing (Collapsing). Using nc distinguished elements orconstants (or nc additional 0-ary operations μ(ci)0 , i = 1, . . . nc),one can decrease arity from n to n′red (as in (3.2)), such thata

μ′n′ = 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 doneby combining iterating and reducing (maybe with additionaloperations of different arity).

aIn DUDEK AND MICHALSKI [1984] μ(c1...cnc )n is named a retract (which term is

already busy and widely used in category theory for another construction).

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`μn

[g`μ(n−1)+1

]. (5.2)

12


Example 5.3. Consider a polyadic version of the binary q-additionwhich 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 6= 1 or ~ 6= 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 `μthat

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`μn [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 apolyadic system is a distinguished element ε (and thecorresponding 0-ary operation μ(ε)0 ) such that for any elementg ∈ 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 inpolyadic systems, there exist many neutral polyads n consisting ofelements of G satisfying

μn [g,n] = g, (5.10)

where g can be also on any place. The neutral polyads are notdetermined 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 oftwo n-ary multiplications

μ2n [g,h,u] = μn [g, μn [h] ,u] = inv. (5.12)

Informally, “internal brackets/multiplication can be moved on anyplace”, 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].

16


Definition 5.8. A polyadic semigroup (n-ary semigroup) is a n-ary

system in which the operation is associative, or

Gsemigrpn = 〈G | μn | associativity〉.

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.

μ2n [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 associativefinal operation μ′n′ during the change of arity.

Example 5.10. An associativity preserving reduction can be givenby 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 aritiesand places were considered in DUDEK AND MICHALSKI [1984],MICHALSKI [1981], SOKHATSKY [1997].

Definition 5.11. An associative polyadic system with identity (5.9)is called a polyadic monoid.

The structure of any polyadic monoid is fixed POP AND POP [2004]:iterating a binary operation CUPONA AND TRPENOVSKI [1961].

18


Several analogs of binary commutativity of polyadic system.

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 polyadicsystem 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. Sofor a n-ary semicommutative system we have

μn[g, hn−1

]= μn

[hn−1, g

]. (5.18)

If a n-ary semigroup Gsemigrp is iterated from a commutative binarysemigroup with identity, then Gsemigrp is semicommutative.

19


Another generalization of commutativity - to generalize binary

mediality in semigroups

(g11 ∙ g12) ∙ (g21 ∙ g22) = (g11 ∙ g21) ∙ (g12 ∙ g22) , (5.19)

following from binary commutativity. For n-ary case they’re different.

Definition 5.12. A polyadic system is medial (or entropic), if

EVANS [1963], BELOUSOV [1972]

μn

μn [g11, . . . , g1n]...

μn [gn1, . . . , gnn]

= μn

μn [g11, . . . , gn1]...

μn [g1n, . . . , gnn]

. (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 isone-to-one in each variable. If g, h are on the same i-th place onboth 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 theequation (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 forall places i is called a n-ary (or polyadic)quasigroup.

Definition 5.16. An associative polyadic quasigroup is called an-ary (or polyadic)group.

21


In a polyadic group the only solution of (5.22) is called aquerelement of g and denoted by g DORNTE [1929], such that

μn [h, g] = g, (5.23)

where g can be on any place. Any idempotent g coincides with itsquerelement g = g. It follows from (5.23) and (5.10), that the polyad

ng =(gn−2g

)(5.24)

is neutral for any element of a polyadic group, where g can be onany 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 andlast places DORNTE [1929], TIMM [1972], or on some other 2places ). In a polyadic group the Dornte 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 unaryqueroperation

μ1 [g] = g. (5.26)

Definition 5.17. A polyadic group is a universal algebra

Ggrpn = 〈G | μn, μ1 | associativity, Dornte relations〉 , (5.27)

where μn is n-ary associative operation and μ1 is thequeroperation (5.26), such that the following diagram

G×(n)id×(n−1) ×μ1 G×n

μ1×id×(n−1)

G×n

G×G

id×Diag(n−1)Pr1

G

μnPr2

G×G

Diag(n−1)×id

(5.28)

commutes, where μ1 can be only on the first and second placesfrom 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`μn (4.1) (cf. DUDEK [1980] for `μ = 2 and GAL’MAK

[2007]).

Definition 5.18. Let us define the querpower k of g recursively

g〈〈k〉〉 =(g〈〈k−1〉〉

), (5.29)

where g〈〈0〉〉 = g, g〈〈1〉〉 = g, or as the k composition

μ◦k1 =

k︷︸︸︷μ1 ◦ μ1 ◦ . . . ◦ μ1 of the queroperation (5.26).

For instance, μ◦21 = μn−3n , such that for any ternary group μ◦21 = id,

i.e. one has g = 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`μn

[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 − 1q − 1

, (5.31)

which have the “nondeformed” limit q → 1 as [k]q → k. Then

g〈〈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 G′n′ = 〈G′ | μ′n′〉 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 G′n. 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 G′n, if

ϕGG′

n+1 (μn [g1, . . . , gn]) = μ′n

[ϕGG

′

1 (g1) , . . . , ϕGG′

n (gn)], gi ∈ G.

(6.1)

26


In general, one should add to this definition the “mapping” of themultiplications

μnψ(μμ′)nn′7→ μ′n′ . (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

GϕGG

′

n+1G′

...............ψ(μ)nn ................

G×n

μnϕGG

′

1 ×...×ϕGG′

n (G′)×n

μ′n (6.3)

If all the components ϕGG′

i of a homotopy are bijections, it is calledan isotopy. In case of polyadic quasigroups BELOUSOV [1972] allmappings ϕGG

′

i are usually taken as permutations of the sameunderlying set G = G′.

27


If the multiplications are also coincide μn = μ′n, then the set

{ϕGGi ; id

}is called an autotopy of the polyadic system Gn.

The diagonal counterparts of homotopy, isotopy and autotopy

(when all mappings ϕGGi coincide) are homomorphism,

isomorphism and automorphism.

A homomorphism from Gn to G′n is given, if there exists one

mapping ϕGG′: G→ G′ satisfying

ϕGG′

(μn [g1, . . . , gn]) = μ′n

[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 Hosszu-Gluskin theorem


Consider concrete forms of polyadic multiplication in terms oflesser arity operations.

History. Simplest way of constructing a n-ary product μ′n from thebinary one μ2 = (∗) is `μ = n iteration (4.1) SUSCHKEWITSCH

[1935], MILLER [1935]

μ′n [g] = g1 ∗ g2 ∗ . . . ∗ gn, gi ∈ G. (7.1)

In DORNTE [1929] it was noted that not all n-ary groups have aproduct of this special form.

The binary group G∗2 = 〈G | μ2 = ∗, e〉 was called a covering groupof the n-ary group G′n = 〈G | μ

′n〉 in POST [1940] (also, TVERMOES

[1953]), where a theorem establishing a more general (than (7.1))structure of μ′n [g] in terms of subgroup structure was given.

29


A manifest form of the n-ary group product μ′n [g] in terms of the

binary one and a special mapping was found in HOSSZU [1963],

GLUSKIN [1965] and is called the Hosszu-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

μ′n [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+1G×(n+1)

G×(n)

ε

μ(e)n G

μ×n2 (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 μnis (see (5.15))

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 polyadicsystem 〈G | μn〉 can be distinguished and may serve as the identityof 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 andis 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, g

)and the binary group

inverse g−1 is g−1 = μn[e, gn−3, g, e

], the polyadic Maltsev term

becomes SHCHUCHKIN [2003]

p (g, e, h) = μn[g, en−3, e, h

]. (7.11)

33


Derive the Hosszu-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 (e, e) or its powers(ek, ek

). Thus, taking for all insertions the minimal number of

neutral polyads, we get

μ3 [g, h, u] = μ73

g,

∗

↓e , e, h, e,

∗

↓e , e, e, u, e, e,

∗

↓e , e, e, e

. (7.13)

34


We rewrite (7.13) as

μ3 [g, h, u] = μ33

g,

∗

↓e , μ3 [e, h, e] ,

∗

↓e ,μ23 [e, e, u, e, e] ,

∗

↓e , 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, e] . (7.19)

35


Thus, we get the Hosszu-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〉 = μk3[e2k+1

]of the binary

identity e are obviously also fixed points ϕ(e〈k〉

)= e〈k〉.

By analogy, the Hosszu-Gluskin “chain formula” for arbitrary n can

be obtained using substitution e→ 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 Hosszu-Gluskin “chain formula” for

arbitrary n

μn [g1, . . . , gn] = g1∗ϕ (g2)∗ϕ2 (g3)∗. . .∗ϕ

n−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 Hosszu-Gluskin theorem in the language of

polyadic powers.

Theorem 7.2. On a polyadic group Gn = 〈G | μn, μ1〉 one can

define a binary group G∗2 = 〈G | μ2 = ∗, e〉 and its automorphism ϕ

such that the Hosszu-Gluskin “chain formula” (7.23) is valid.

37


The following reverse Hosszu-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)∗ . . .∗ϕ

n−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 Hosszu-Gluskin chain formula


We can generalize the Hosszu-Gluskin chain formula, if the number

of the inserted neutral polyads can be chosen arbitrarily, not only

minimally, as they are neutral. Indeed, in the particular case n = 3,

we put the map ϕ as

ϕq (g) = μ`ϕ(q)3 [e, g, eq] , (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, e, (e, h, eq) , e, (e, u, eq)

q+1, e, eq(q+1)+1

]. (8.3)

39


Therefore we have obtained the “q-deformed” homotopy maps

ψ1 (g) = ϕ[[0]]qq (g) = ϕ0q (g) = g, (8.4)

ψ2 (g) = ϕq (g) = ϕ[[1]]qq (g) , (8.5)

ψ3 (g) = ϕq+1q (g) = ϕ

[[2]]qq (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 andwe put ϕ0q = id. The corresponding “q-deformed” chain formula (forn = 3) can be written as

μ3 [g, h, u] = g ∗ ϕ[[1]]qq (h) ∗ ϕ

[[2]]qq (u) ∗ bq, (8.8)

bq = e〈`e(q)〉, (8.9)

`e (q) = q[[2]]q2

. (8.10)

40


The “nondeformed” limit q → 1 of (8.8) gives the standard

Hosszu-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− 1is 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-endomorphismGLUSKIN AND SHVARTS [1972] (which has one multiplication andleads to the standard “nondeformed” chain formula) of the binarygroup 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 [2013b]

ϕq (g) ∗ ϕq (h) = ϕq(g ∗ ϕq (e) ∗ h

). (8.13)

The corresponding diagram

G×Gμ2 G

ϕqG

G×G

ϕq×ϕqεG× {•} ×G

id×μ(e)0 ×id G×G×G

μ2×μ2 (8.14)

commutes. If q = 1, then ϕq (e) = e, and the distinguished elementa 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]]qq (g) = ϕ0q (g) = g, (8.15)

ψ2 (g) = ϕq (g) = ϕ[[1]]qq (g) , (8.16)

ψ3 (g) = ϕq+1q (g) = ϕ

[[2]]qq (g) , (8.17)

...

ψn−1 (g) = ϕqn−3+...+q+1q (g) = ϕ

[[n−2]]qq (g) , (8.18)

ψn (g) = ϕqn−2+...+q+1q (g) = ϕ

[[n−1]]qq (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]]qn− 1

. (8.21)

43


Thus the “q-deformed” n-ary chain formula is DUPLIJ [2013b]

μn [g1, . . . , gn] = g1∗ϕ[[1]]qq (g2)∗ϕ

[[2]]qq (g3)∗. . .∗ϕ

[[n−2]]qq (gn−1)∗ϕ

[[n−1]]qq (gn)∗e

〈`e(q)〉.

(8.22)

In the “nondeformed” limit q → 1 (8.22) reproduces the standardHosszu-Gluskin chain formula (7.23). Instead of the fixed pointrelation (7.27) we now have the quasi-fixed point

ϕq (bq) = bq ∗ ϕq (e) , (8.23)

where the “deformed” distinguished element bq is

bq = μ•n

[e[[n]]q

]= e〈`e(q)〉. (8.24)

The conjugation relation (7.26) in the “deformed” case becomesthe quasi-conjugation DUPLIJ [2013b]

ϕ[[n−1]]qq (g) ∗ bq = bq ∗ ϕ

[[n−1]]qq (e) ∗ g. (8.25)

44


We formulate the following “q-deformed” analog of the

Hosszu-Gluskin theorem DUPLIJ [2013b].

Theorem 8.1. On a polyadic group Gn = 〈G | μn, μ1〉 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]]qq (g) = bq ∗ ϕ

[[n−1]]qq (e) ∗ g ∗ b−1q . (8.27)

45

9 (One-place) generalizations of homomorphisms


Definition 9.1. The n-ary homomorphism is realized as asequence of n consequent (binary) homomorphisms ϕi,i = 1, . . . , n, of n similar polyadic systems

n︷︸︸︷

Gnϕ1→ G′n

ϕ2→ . . .ϕn−1→ G′′n

ϕn→ G′′′n (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 thesame formula (6.4), where both do not change arity (are equiary);

2) use one modified (and not equiary) diagram and the underlyingformula (6.4) by themselves, which will allow us to change aritywithout 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′ | μ′n′〉 the

following additional term operations of opposite arity

νn′ : G×n′ → G and ν′n : G

′×n → G′ and consider two equiary

mappings between 〈G | μn, νn′〉 and 〈G′ | μ′n′ , ν′n〉.

47


A weak homomorphism from 〈G | μn, νn′〉 to 〈G′ | μ′n′ , ν′n〉 is given,

if there exists a mapping ϕGG′: G→ G′ satisfying two relations

simultaneously

ϕGG′

(μn [g1, . . . , gn]) = ν′n

[ϕGG

′

(g1) , . . . , ϕGG′ (gn)

], (9.2)

ϕGG′

(νn′ [g1, . . . , gn′ ]) = μ′n′

[ϕGG

′

(g1) , . . . , ϕGG′ (gn′)

]. (9.3)

GϕGG

′

G′

........ψ(μν′)

nn.........

G×n

μn (ϕGG

′)×n

(G′)×n

ν′n

GϕGG

′

G′

..........ψ(νμ′)

n′n′...........

G×n′

νn′(ϕGG

′)×n′

(G′)×n′

μ′n′ (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 (ìd-intact) id-product for the n-ary system 〈G | μn〉 as

μ(ìd)n = μn × (idG)×ìd , (10.1)

μ(ìd)n : G×(n+ìd) → G×(1+ìd). (10.2)

To indicate the exact i-th place of μn in (10.1), we write μ(ìd)n (i).

Introduce a multiplace mapping Φ(n,n′)k acting as DUPLIJ [2013a]

Φ(n,n′)k : G×k → G′. (10.3)

49


We have the following commutative diagram which changes arity

G×kΦk

G′

G×kn

μ(`id)n

(Φk)×n′

(G′)×n′μ′n′ (10.4)

Definition 10.1. A k-place heteromorphism from Gn to G′n′ 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 [2013a]

Φ(n,n′)k

μn [g1, . . . , gn]

gn+1...

gn+`id

= μ′n′

Φ(n,n′)k

g1...

gk

, . . . ,Φ

(n,n′)k

gk(n′−1)...

gkn′

.

(10.5)

The notion of heteromorhism is motivated by ELLERMAN [2006,

2007], where mappings between objects from different categories

were considered and called ’chimera morphisms’.

51


In the particular case n = 3, n′ = 2, k = 2, `id = 1 we have

Φ(3,2)2

μ3 [g1, g2, g3]

g4

= μ′2

Φ(3,2)2

g1

g2

,Φ(3,2)2

g3

g4

.

(10.6)

This was used in the construction of the bi-element representations

of ternary groups BOROWIEC ET AL. [2006], DUPLIJ [2013a].

Example 10.2. Let G =Madiag2 (K), a set of antidiagonal 2× 2

matrices over the field K and G′ = K, where K = R,C,Q,H. The

ternary multiplication μ3 is a product of 3 matrices. Obviously, μ3 is

nonderived.

52


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. So we can have the relationbetween the heteromorhism Φ(3,2)2 and the homomorphism 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.

53


For the noncommutative field K (= Q or H) we can define the

heteromorphism (10.7) only.

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 ìd-intact id-product for 〈G;μn〉 as

μ(`μ,ìd)n = (μn)×`μ × (idG)

×ìd , (10.9)

μ(`μ,ìd)n : G×(n`μ+ìd) → G×(`μ+ìd). (10.10)

A `μ-ple k-place heteromorphism from Gn to G′n′ is given, if there

exists a k-place mapping Φ(n,n′)k (10.3).

54


The main heteromorphism equation is DUPLIJ [2013a]

Φ

(n,n′

)

k

μn [g1, . . . , gn] ,

.

.

.

μn

[gn(`μ−1

), . . . , gn`μ

]

`μ

gn`μ+1,

.

.

.

gn`μ+ìd

ìd

= μ′n′

Φ

(n,n′

)

k

g1

.

.

.

gk

, . . . ,Φ

(n,n′

)

k

gk(n′−1

)

.

.

.

gkn′

.

(10.11)

It is a polyadic analog of Φ(g1 ∗ g2) = Φ (g1) • Φ(g2), whichcorresponds to n = 2, n′ = 2, k = 1, `μ = 1, ìd = 0, μ2 = ∗, μ

′2 = •.

We obtain two arity changing formulas

n′ = n−n− 1k

ìd, (10.12)

n′ =n− 1k

`μ + 1, (10.13)

where n−1kìd ≥ 1 and n−1

k`μ ≥ 1 are integer.

55


The following inequalities hold valid

1 ≤ `μ ≤ k, (10.14)

0 ≤ ìd ≤ 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 ìd 6= 0 then it is change of the arity n′ 6= n.

If ìd = 0, then k = kmin = `μ, and no change of arity n′max = n.

We call a heteromorphism having ìd = 0 a k-place homomorphismwith k = `μ. An opposite extreme case, when the final arityapproaches its minimum n′min = 2 (the final operation is binary),corresponds to the maximal value of places k = kmax = (n− 1) `μ.

56


Рис. 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 ìd = const (ìd = 1 (solid), ìd = 2 (dash));

right: fixed multiplications `μ = const (`μ = 1 (solid), `μ = 2 (dash)).

57


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′ = n′max = 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− 1n′ − 1

`μ, `id =n− n′

n′ − 1`μ. (10.18)

3. n′ = n′min = 2 =⇒ Φ(n,2)(n−1)`μ

is the (n− 1) `μ-place binarizing

heteromorphism, i.e., k = kmax = (n− 1) `μ.

58


Таблица 1: “Quantization” of heteromorphisms

k `μ `id n/n′

2 1 1n = 3, 5, 7, . . .

n′ = 2, 3, 4, . . .

3 1 2n = 4, 7, 10, . . .

n′ = 2, 3, 4, . . .

3 2 1n = 4, 7, 10, . . .

n′ = 3, 5, 7, . . .

4 1 3n = 5, 9, 13, . . .

n′ = 2, 3, 4, . . .

4 2 2n = 3, 5, 7, . . .

n′ = 2, 3, 4, . . .

4 3 1n = 5, 9, 13, . . .

n′ = 4, 7, 10, . . .59

11 Associativity, quivers and heteromorphisms


Semigroup heteromorphisms: associativity of the final operation

μ′n′ , 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 μnDUPLIJ [2013a]. 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).

60


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 is

Φ(4,2)3

μ4 [g1, h2, g2, u1]

h1

u2

= μ

′2

Φ(4,2)3

g1

h1

u1

,Φ

(4,2)3

g2

h2

u2

.

(11.1)

As it is seen from (11.1), the product μ′2 is not associative, if μ4 is

associative.

Definition 11.1. An associative polyadic quiver is a polyadic quiver

which ensures the final associativity of μ′n′ in the main

heteromorphism equation (10.11), when the initial multiplication μnis associative.

61


One of the associative polyadic quivers which corresponds to thesame 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

= μ′2

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 therules of construction of corresponding heteromorphism equations,i.e. consistent procedure for building semigroup heteromorphisms.

62


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]

= μ′3

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]

= μ′3

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 ET AL. [2006], DUPLIJ [2013a].

63


An important class of intactless heteromorphisms (with `id = 0)preserving associativity can be constructed using an analogy withthe Post substitutions POST [1940], and therefore we call it thePost-like associative quiver. The number of places k is now fixed byk = n− 1, while n′ = n and `μ = k = n− 1. An example of thePost-like associative quiver with the same heteromorphismsparameters as in (11.3)-(11.4) is

Φ(3,3)2

μ3 [g1, h2, g3]

μ3 [h1, g2, h3]

= μ′3

Φ(3,3)2

g1

h1

,Φ(3,3)2

g2

h2

,Φ(3,3)2

g3

h3

.

(11.5)This construction appeared in the study of ternary semigroups ofmorphisms CHRONOWSKI [1994]. Its n-ary generalization was usedspecial representations of n-groups GLEICHGEWICHT ET AL. [1983],WANKE-JAKUBOWSKA AND WANKE-JERIE [1984] (where the n-groupwith the multiplication μ′2 was called the diagonal n-group).

64

12 Multiplace representations of polyadic systems


The final multiplication μ′n′ is a linear map, which leads to

restrictions on the final polyadic structure G′n′ .

Let V be a vector space over a field K and EndV 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 → EndV , such that Π1is a (one-place) homomorphism

Π1 (μ2 [g, h]) = Π1 (g) ∗Π1 (h) , (12.1)

where g, h ∈ G and (∗) is the binary multiplication in EndV .

65


If G2 is a binary group with the unity e, then we have the additional

condition

Π1 (e) = idV . (12.2)

General idea: to use the heteromorphism equation (10.11) instead

of the standard homomorphism equation (12.1), such that the arity

of the representation will be different from the arity of the initial

polyadic system n′ 6= n.

Consider the structure of the final n′-ary multiplication μ′n′ in

(10.11), taking into account that the final polyadic system G′n′should be constructed from EndV . The most natural and physically

applicable way is to consider the binary EndV and to put

G′n′ = dern′ (EndV ), as it was proposed for the ternary case in

BOROWIEC ET AL. [2006], DUPLIJ [2013a].

66


In this way G′n′ becomes a derived n′-ary (semi)group of

endomorphisms of V with the multiplication μ′n′ :

(EndV )×n′ → EndV , where

μ′n′ [v1, . . . , vn′ ] = v1 ∗ . . . ∗ vn′ , vi ∈ EndV. (12.3)

Because the multiplication μ′n′ (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 → EndV. (12.4)

67


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) which satisfies the (associativity preserving)heteromorphism equation (10.11), that is DUPLIJ [2013a]

Π

(n,n′

)

k

μn [g1, . . . , gn] ,

.

.

.

μn

[gn(`μ−1

), . . . , gn`μ

]

`μ

gn`μ+1,

.

.

.

gn`μ+ìd

ìd

=

n′︷︸︸︷

Π

(n,n′

)

k

g1

.

.

.

gk

∗ . . . ∗ Π

(n,n′

)

k

gk(n′−1

)

.

.

.

gkn′

,

(12.5)

G×kΠk

EndV

G×kn′

μ(`μ,ìd)n

(Πk)×n′

(EndV )×n′

(∗)n′ (12.6)

where μ(`μ,ìd)n is `μ-ple ìd-intact id-product given by (10.9).

68


General classification of multiplace representations can be done byanalogy with that of the heteromorphisms as follows:

1. The hom-like multiplace representation which is a multiplacehomomorphism with n′ = n′max = 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 theintermediate heteromorphism with 2 < n′ < n and the numberof intact elements is

lid =n− n′

n′ − 1`μ. (12.7)

3. The binary multiplace representation which is a binarizingheteromorphism (3) with n′ = n′min = 2, the maximal number of

intact elements l(max)id = (n− 2) `μ and maximal number ofplaces

k = kmax = (n− 1) `μ. (12.8)

69


In case of n-ary groups, we need an analog of the “normalizing”

relation (12.2). If the n-ary group has the unity e, then

Π(n,n′)k

e

...

e

k

= idV . (12.9)

70


If there is no unity at all, one can “normalize” the multiplacerepresentation, using analogy with (12.2) in the form

Π1(h−1 ∗ h

)= idV , (12.10)

as follows

Π(n,n′)k

h

...

h

`μ

h

...

h

`id

= idV , (12.11)

for all h ∈Gn, where h is the querelement of h. The latter ones canbe placed on any places in the l.h.s. of (12.11) due to the Dornteidentities.

71


A general form of multiplace representations can be found by

applying the Dornte 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, g] , a = 1, . . . , `μ. (12.13)

72

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 operationρ(n)1 : G× X→ X such that it is consistent with the multiplicationμn, i.e. composition of the binary operations ρ1 {g|x} gives then-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) itis implied the there exist such ex ∈ G (which may or may notcoincide with the unity of Gn) that ρ(n)1 {ex|x} = x for all x ∈ X, and

the mapping x 7→ ρ(n)1 {ex|x} is a bijection of X. The right 1-placeactions of Gn on X are defined in a symmetric way, and thereforewe will consider below only one of them.

73


Obviously, we cannot compose ρ(n)1 and ρ(n′)

1 with n 6= n′. Usually

X is called a G-set or G-space depending on its properties (see,

e.g., HUSEMOLLER ET AL. [2008]).

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).

74


We propose (schematically) DUPLIJ [2013a]

ρ(n)k

μn [g1, . . . , gn] ,

.

.

.

μn

[gn(`μ−1

), . . . , gn`μ

]

`μ

gn`μ+1,

.

.

.

gn`μ+`id

`id

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

x

= ρ(n)k

n′︷︸︸︷

g1

.

.

.

gk

∣∣∣∣∣∣∣∣∣∣∣

. . .

∣∣∣∣∣∣∣∣

ρ(n)k

gk(n′−1

)

.

.

.

gkn′

∣∣∣∣∣∣∣∣∣∣∣

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.

75


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)

76


In terms of the querelement, the normalization has the form

ρ(n)k

h

...

h

`μ

h

...

h

`id

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

x

= x, for all x ∈ X and for all h ∈ Gn.

(13.5)

77


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).

78


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.,

HUSEMOLLER 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).

79


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].

80

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 ρreg(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.

81


Example 14.1. Let G3 be a ternary semigroup, k = 2, and X =G3,

then 2-place (left) action can be defined as

ρreg(3)2

g

h

∣∣∣∣∣∣u

def= μ3 [g, h, u] . (14.1)

This gives the following composition law for two regular multiactions

ρreg(3)2

g1

h1

∣∣∣∣∣∣ρreg(3)2

g2

h2

∣∣∣∣∣∣u

= μ3 [g1, h1, μ3 [g2, h2, u]]

= μ3 [μ3 [g1, h1, g2] , h2, u] = ρreg(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).

82


The formula (14.1) can be simultaneously treated as a 2-translation

BELOUSOV [1972]. In this way, the following left regular multiaction

ρreg(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 n′reg, which

differs for even and odd values of the initial arity n.

83


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

n′reg = n− `id (14.5)

n′reg = `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.

84


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 ET AL. [2006], DUPLIJ [2013a] 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.

85


Example 14.2. Let us consider 5-ary semigroup and the following

middle 4-action

ρreg(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 associative quiver (11.4), but with a more

86


complicated l.h.s., that is

ρreg(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

=

ρreg(5)4

g1

h1

u1

v1

∣∣∣∣∣∣∣∣∣∣∣

ρreg(5)4

g2

h2

u2

v2

∣∣∣∣∣∣∣∣∣∣∣

ρreg(5)4

g3

h3

u3

v3

∣∣∣∣∣∣∣∣∣∣∣

ρreg(5)4

g4

h4

u4

v4

∣∣∣∣∣∣∣∣∣∣∣

ρreg(5)4

g5

h5

u5

v5

∣∣∣∣∣∣∣∣∣∣∣

s

.

(14.8)

87


Now we have an additional case with two intact elements `id and

two multiplications `μ = 2 as

ρreg(5)4

μ5 [g1, h1, g2, h2,g3]

h3

μ5 [h3, v3, u2, v2, u1]

v1

∣∣∣∣∣∣∣∣∣∣∣

s

= ρreg(5)4

g1

h1

u1

v1

∣∣∣∣∣∣∣∣∣∣∣

ρreg(5)4

g2

h2

u2

v2

∣∣∣∣∣∣∣∣∣∣∣

ρreg(5)4

g3

h3

u3

v3

∣∣∣∣∣∣∣∣∣∣∣

s

,

(14.9)

with arity changing from n = 5 to n′reg = 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.

88


For n > 3, in addition to left, right and middle multiactions, there

exist intermediate cases. First, observe that the i-translations with

i = 2 and i = n− 1 immediately fix the final arity n′reg = n.

Therefore, the composition of multiactions will be similar to (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 ZEKOVIC AND ARTAMONOV [1999, 2002]).

89


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 [2013a]

Πreg(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

ET AL. [2006], DUPLIJ [2013a] for ternary groups.

90

15 Matrix representations of ternary groups


Here we give several examples of matrix representations for

concrete ternary groups BOROWIEC ET AL. [2006], DUPLIJ [2013a].

Let G = Z3 3 {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 manifestmatrix form

ΠLreg (0, 0) = Π

Lreg (2, 2) = Π

Lreg (1, 1) = Π

Rreg (0, 0)

= ΠRreg (2, 2) = Π

Rreg (1, 1) =

1 0 0

0 1 0

0 0 1

= [1] ⊕ [1] ⊕ [1], (15.2)

91


ΠLreg (2, 0) = Π

Lreg (1, 2) = Π

Lreg (0, 1) = Π

Rreg (2, 1) = Π

Rreg (1, 0) = Π

Rreg (0, 2) =

0 1 0

0 0 1

1 0 0

= [1] ⊕

−1

2−

√3

2√3

2−1

2

= [1] ⊕

[−1

2+1

2i√3

]⊕[−1

2−1

2i√3

], (15.3)

ΠLreg (2, 1) = Π

Lreg (1, 0) = Π

Lreg (0, 2) = Π

Rreg (2, 0) = Π

Rreg (1, 2) = Π

Rreg (0, 1) =

0 0 1

1 0 0

0 1 0

= [1] ⊕

−1

2

√3

2

−

√3

2−1

2

= [1] ⊕

[−1

2−1

2i√3

]⊕[−1

2+1

2i√3

]. (15.4)

92


Consider next the middle representation construction. The middle

regular representation is defined by

ΠMreg (g1, g2) t =

n∑

i=1

ki [g1hig2] .

For regular representations we have

ΠMreg (g1, h1) ◦ΠRreg (g2, h2) = Π

Rreg (h2, h1) ◦Π

Mreg (g1, g2) , (15.5)

ΠMreg (g1, h1) ◦ΠLreg (g2, h2) = Π

Lreg (g1, g2) ◦Π

Mreg (h2, h1) . (15.6)

93


For the middle regular representation matrices we obtain

ΠMreg (0, 0) = ΠMreg (1, 2) = Π

Mreg (2, 1) =

1 0 0

0 0 1

0 1 0

,

ΠMreg (0, 1) = ΠMreg (1, 0) = Π

Mreg (2, 2) =

0 1 0

1 0 0

0 0 1

,

ΠMreg (0, 2) = ΠMreg (2, 0) = Π

Mreg (1, 1) =

0 0 1

0 1 0

1 0 0

.

The above representation ΠMreg of 〈Z3, [ ]〉 is equivalent to the

94


orthogonal direct sum of two irreducible representations

ΠMreg (0, 0) = ΠMreg (1, 2) = Π

Mreg (2, 1) = [1]⊕

−1 0

0 1

,

ΠMreg (0, 1) = ΠMreg (1, 0) = Π

Mreg (2, 2) = [1]⊕

1

2−

√3

2

−

√3

2−1

2

,

ΠMreg (0, 2) = ΠMreg (2, 0) = Π

Mreg (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) 6= idV , but

ΠM (g, h) ◦ΠM (g, h) = idV , and so ΠM are of the second degree.

95


Consider a more complicated example of left representations. Let

G = Z4 3 {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

96


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

ΠLreg (g1, g2) t =∑ni=1 ki [g1g2hi], which means that (see the

general formula (14.10))

ΠLreg (g, h) |u >= | [ghu] > .

Analogously, for right and middle representations

ΠRreg (g, h) |u >= | [ugh] >, ΠMreg (g, h) |u >= | [guh] > .

Therefore | [ghu] >= | [ugh] >= | [guh] > and

ΠLreg (g, h) = ΠRreg (g, h) |u >= Π

Mreg (g, h) |u >,

so ΠLreg (g, h) = ΠRreg (g, h) = Π

Mreg (g, h). Thus it is sufficient to

consider the left representation only.

97


In this case the equivalence isΠL(a, b) = ΠL(c, d)⇐⇒ (a+ b) = (c+ d)mod 4, and we obtain thefollowing classes

ΠLreg (0, 0) = Π

Lreg (1, 3) = Π

Lreg (2, 2) = Π

Lreg (3, 1) =

0 0 0 1

1 0 0 0

0 1 0 0

0 0 1 0

= [1] ⊕ [−1] ⊕ [−i] ⊕ [i] ,

ΠLreg (0, 1) = Π

Lreg (1, 0) = Π

Lreg (2, 3) = Π

Lreg (3, 2) =

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0

= [1] ⊕ [−1] ⊕ [−1] ⊕ [−1] ,

ΠLreg (0, 2) = Π

Lreg (1, 1) = Π

Lreg (2, 0) = Π

Lreg (3, 3) =

0 1 0 0

0 0 1 0

0 0 0 1

1 0 0 0

= [1] ⊕ [−1] ⊕ [i] ⊕ [−i] ,

ΠLreg (0, 3) = Π

Lreg (1, 2) = Π

Lreg (2, 1) = Π

Lreg (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.

98


Let us “algebralize” the regular representations DUPLIJ [2013a].From (7.15) we have, for the left representation

ΠLreg (i, j) ◦ΠLreg (k, l) = Π

Lreg (i, [jkl]) , (15.8)

where [jkl] = j − k + l, i, j, k, l ∈ Z3. Denote γLi = ΠLreg (0, i),

i ∈ Z3, then we obtain the algebra with the relations

γLi γLj = γ

Li+j . (15.9)

Conversely, any matrix representation of γiγj = γi+j leads to theleft representation by ΠL (i, j) = γj−i. In the case of the middleregular representation we introduce γMk+l = Π

Mreg (k, l), k, l ∈ Z3,

then we obtain

γMi γMj γ

Mk = γ

M[ijk], i, j, k ∈ Z3. (15.10)

In some sense (15.10) can be treated as a ternary analog of theClifford algebra. Now the middle representation is ΠM (k, l) = γk+l.

99


Acknowledgements

I am deeply thankful to A. Borowiec, L. Carbone, D. Ellerman,

W. Dudek, A. Galmak, M. Gerstenhaber, S. Grigoryan,

V. Khodusov, G. Kurinnoy, M. Putz, T. Nordahl, B. Novikov,

Ya. Radyno, V. Retakh, B. Schein, J. Stasheff, E.Taft, R. Umble,

A. Vershik, G. Vorobiev, A. Voronov, W. Werner, and C. Zachos for

numerous fruitful discussions.

100

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