Polyadic systems and multiplace representations

Polyadic Systems and
Multiplace Representations
STEVEN DUPLIJ
Bochum & M¨unster
http://wwwmath.uni-muenster.de/u/duplij
2016
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 DÖRNTE
[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]. Some theorems connecting representations of binary and
n-ary groups were presented in DUDEK AND SHAHRYARI [2012].
Ternary fields were developed in DUPLIJ AND WERNER [2015].
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.
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)
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). 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 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 | μn 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 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 Preliminaries
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 thata
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).
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 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].
16

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 .
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].
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 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 DÖRNTE [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 DÖRNTE [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
Ggrp
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 .
27

If the multiplications are also coincide μn = μn, then the set
ϕGG
i ; id is called an autotopy of the polyadic system Gn.
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. We take the
number of the inserted neutral polyads 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, 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 [2013]
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 [2013]
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 [2013]
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 [2013].
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)
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

 = 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].
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. 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 relation
between 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 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).
54

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.
55

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.
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 id = const ( id = 1 (solid), id = 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 = 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.
58

Таблица 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, . . .59

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).
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 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.
61

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.
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]

 = 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].
63

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).
64

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 .
65

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].
66

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)
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) 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).
68

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)
69

In case of n-ary groups, we need an analog of the “normalizing”
relation (12.2) Π1 (e) = idV . 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 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.
71

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)
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 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.
73

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).
74

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.
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



ˉh
...
ˉh



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.,
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).
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 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.
81

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).
82

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.
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
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.
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, 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.
85

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 associative quiver (11.4), but with a more
86

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)
87

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.
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 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]).
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 [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.
90

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)
91

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)
92

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)
93

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
94

orthogonal direct 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.
95

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







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
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.
97

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.
98

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.
99

Acknowledgements
I am deeply thankful to A. Borowiec, L. Carbone, J.Cuntz, D.
Ellerman, W. Dudek, A. Galmak, M. Gerstenhaber, S. Grigoryan,
V. Khodusov, G. Kurinnoy, M. P¨utz, T. Nordahl, B. Novikov, N.C.
Phillips, 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

Bibliography
Bibliography
ABRAMOV, V. (1995). Z3-graded analogues of Clifford algebras and
algebra of Z3-graded symmetries. Algebras Groups Geom. 12
(3), 201–221.
ABRAMOV, V. (1996). Ternary generalizations of Grassmann
algebra. Proc. Est. Acad. Sci., Phys. Math. 45 (2-3), 152–160.
ABRAMOV, V., R. KERNER, AND B. LE ROY (1997). Hypersymmetry: a
Z3-graded generalization of supersymmetry. J. Math. Phys. 38,
1650–1669.
BAGGER, J. AND N. LAMBERT (2008a). Comments on multiple
M2-branes. J. High Energy Phys. 2, 105.
BAGGER, J. AND N. LAMBERT (2008b). Gauge symmetry and
supersymmetry of multiple M2-branes. Phys. Rev. D77, 065008.
101

Bibliography
BELOUSOV, V. D. (1972). n-ary Quasigroups. Kishinev: Shtintsa.
BERGMAN, C. (2012). Universal Algebra: Fundamentals and
Selected Topics. New York: CRC Press.
BERGMAN, G. M. (1995). An invitation to general algebra and
universal constructions. Berkeley: University of California.
BOROWIEC, A., W. DUDEK, AND S. DUPLIJ (2006). Bi-element
representations of ternary groups. Comm. Algebra 34 (5),
1651–1670.
CHRONOWSKI, A. (1994). A ternary semigroup of mappings.
Demonstr. Math. 27 (3-4), 781–791.
ˇCUPONA, G. AND B. TRPENOVSKI (1961). Finitary associative
operations with neutral elements. Bull. Soc. Math. Phys.
Macedoine 12, 15–24.
102

Bibliography
DE AZCARRAGA, J. A. AND J. M. IZQUIERDO (2010). n-Ary algebras: A
review with applications. J. Phys. A43, 293001.
DENECKE, K. AND K. SAENGSURA (2008). State-based systems and
(F1, F2)-coalgebras. East-West J. Math., Spec. Vol., 1–31.
DENECKE, K. AND S. L. WISMATH (2009). Universal Algebra and
Coalgebra. Singapore: World Scientific.
D¨ORNTE, W. (1929). Unterschungen ¨uber einen verallgemeinerten
Gruppenbegriff. Math. Z. 29, 1–19.
DUDEK, W. A. (1980). Remarks on n-groups. Demonstratio
Math. 13, 165–181.
DUDEK, W. A. (2007). Remarks to Głazek’s results on n-ary groups.
Discuss. Math., Gen. Algebra Appl. 27 (2), 199–233.
DUDEK, W. A. AND J. MICHALSKI (1984). On retracts of polyadic
groups. Demonstratio Math. 17, 281–301.
103

Bibliography
DUDEK, W. A. AND M. SHAHRYARI (2012). Representation theory of
polyadic groups. Algebr. Represent. Theory 15 (1), 29–51.
DUPLIJ, S. (2012). Polyadic systems, representations and quantum
groups. J. Kharkov National Univ., ser. Nuclei, Particles and
Fields 1017 (3(55)), 28–59. (for considerably extended version
with diagrams, tables, figures, see arXiv:1308.4060).
DUPLIJ, S. (2013). A “q-deformed” generalization of the
Hosszú-Gluskin theorem, preprint Mathematisches Institut,
Münster, 18 p., arXiv:math.RA/1309.1134v2 (To appear in
Filomat, 2016).
DUPLIJ, S. AND W. WERNER (2015). Structure of unital 3-fields,
preprint Mathematisches Institut, Münster, 18 p.,
arXiv:1505.04393 (To appear in J. Algebra, 2016).
104

Bibliography
ELLERMAN, D. (2006). A theory of adjoint functors – with some
thoughts about their philosophical significance. In: G. SICA (Ed.),
What is category theory?, Monza (Milan): Polimetrica, pp.
127–183.
ELLERMAN, D. (2007). Adjoints and emergence: applications of a
new theory of adjoint functors. Axiomathes 17 (1), 19–39.
EVANS, T. (1963). Abstract mean values. Duke Math J. 30,
331–347.
FILIPPOV, V. T. (1985). n-Lie algebras. Sib. Math. J. 26, 879–891.
GAL’MAK, A. M. (1986). Translations of n-ary groups. Dokl. Akad.
Nauk BSSR 30, 677–680.
GAL’MAK, A. M. (1998). Generalized morphisms of algebraic
systems. Vopr. Algebry 12, 36–46.
105

Bibliography
GAL’MAK, A. M. (2001). Polyadic analogs of the Cayley and Birkhoff
theorems. Russ. Math. 45 (2), 10–15.
GAL’MAK, A. M. (2007). n-Ary Groups, Part 2. Minsk: Belarus State
University.
GAL’MAK, A. M. AND G. N. VOROBIEV (2013). On
Post-Gluskin-Hosszu theorem. Probl. Fiz. Mat. Tekh. 2011
(1(14)), 55–60.
GEORGI, H. (1999). Lie Algebras in Particle Physics. New York:
Perseus Books.
GŁAZEK, K. (1980). Weak homomorphisms of general algebras and
related topics. Math. Semin. Notes, Kobe Univ. 8, 1–36.
GŁAZEK, K. AND B. GLEICHGEWICHT (1982). Abelian n-groups. In:
Colloq. Math. Soc. J. Bolyai. Universal Algebra (Esztergom,
1977), Amsterdam: North-Holland, pp. 321–329.
106

Bibliography
GŁAZEK, K. AND J. MICHALSKI (1974). On weak homomorphisms of
general non-indexed algebras. Bull. Acad. Pol. Sci., S´er. Sci.
Math. Astron. Phys. 22, 651–656.
GLEICHGEWICHT, B., M. B. WANKE-JAKUBOWSKA, AND M. E.
WANKE-JERIE (1983). On representations of cyclic n-groups.
Demonstr. Math. 16, 357–365.
GLUSKIN, L. M. (1965). Position operatives. Math. Sbornik 68(110)
(3), 444–472.
GLUSKIN, L. M. AND V. Y. SHVARTS (1972). Towards a theory of
associatives. Math. Notes 11 (5), 332–337.
GOETZ, A. (1966). On weak isomorphisms and weak
homomorphisms of abstract algebras. Colloq. Math. 14,
163–167.
107

Bibliography
GUSTAVSSON, A. (2009). One-loop corrections to Bagger-Lambert
theory. Nucl. Phys. B807, 315–333.
HEINE, E. (1878). Handbuch der Kugelfunktionan. Berlin: Reimer.
HOBBY, D. AND R. MCKENZIE (1988). The Structure of Finite
Algebras. Providence: AMS.
HOSSZ Ú, L. M. (1963). On the explicit form of n-group operation.
Publ. Math. Debrecen 10, 88–92.
HUSEM ÖLLER, D., M. JOACHIM, B. JUR ˇCO, AND M. SCHOTTENLOHER
(2008). G-spaces, G-bundles, and G-vector bundles. In:
D. HUSEM ÖLLER AND S. ECHTERHOFF (Eds.), Basic Bundle Theory
and K-Cohomology Invariants, Berlin-Heidelberg: Springer, pp.
149–161.
KAC, V. AND P. CHEUNG (2002). Quantum calculus. New York:
Springer.
108

Bibliography
KAPRANOV, M., I. M. GELFAND, AND A. ZELEVINSKII (1994).
Discriminants, Resultants and Multidimensional Determinants.
Berlin: Birkh¨auser.
KAWAMURA, Y. (2003). Cubic matrices, generalized spin algebra
and uncertainty relation. Progr. Theor. Phys. 110, 579–587.
KIRILLOV, A. A. (1976). Elements of the Theory of Representations.
Berlin: Springer-Verlag.
KOLIBIAR, M. (1984). Weak homomorphisms in some classes of
algebras. Stud. Sci. Math. Hung. 19, 413–420.
L˜OHMUS, J., E. PAAL, AND L. SORGSEPP (1994). Nonassociative
Algebras in Physics. Palm Harbor: Hadronic Press.
MAL’TCEV, A. I. (1954). On general theory of algebraic systems.
Mat. Sb. 35 (1), 3–20.
109

Bibliography
MARCZEWSKI, E. (1966). Independence in abstract algebras.
Results and problems. Colloq. Math. 14, 169–188.
MICHALSKI, J. (1981). Covering k-groups of n-groups. Arch. Math.
(Brno) 17 (4), 207–226.
MILLER, G. A. (1935). Sets of group elements involving only
products of more than n. Proc. Natl. Acad. Sci. USA 21, 45–47.
NAMBU, Y. (1973). Generalized Hamiltonian dynamics. Phys.
Rev. 7, 2405–2412.
NIVANEN, L., A. LE M´EAUT´E, AND Q. A. WANG (2003). Generalized
algebra within a nonextensive statistics. Rep. Math. Phys. 52 (3),
437–444.
POP, M. S. AND A. POP (2004). On some relations on n-monoids.
Carpathian J. Math. 20 (1), 87–94.
110

Bibliography
POST, E. L. (1940). Polyadic groups. Trans. Amer. Math. Soc. 48,
208–350.
RAUSCH DE TRAUBENBERG, M. (2008). Cubic extentions of the
poincare algebra. Phys. Atom. Nucl. 71, 1102–1108.
ROBINSON, D. W. (1958). n-groups with identity elements. Math.
Mag. 31, 255–258.
SHCHUCHKIN, N. A. (2003). An interconnection between n-groups
and groups. Chebyshevski˘ı Sb. 4 (1(5)), 125–141.
SOKHATSKY, F. M. (1997). On the associativity of multiplace
operations. Quasigroups Relat. Syst. 4, 51–66.
SOKOLOV, E. I. (1976). On the theorem of Gluskin-Hossz´u on
D¨ornte groups. Mat. Issled. 39, 187–189.
SOKOLOV, N. P. (1972). Introduction to the Theory of
Multidimensional Matrices. Kiev: Naukova Dumka.
111

Bibliography
SUSCHKEWITSCH, A. (1935). ¨Uber die Erweiterung der Semigruppe
bis zur ganzen Gruppe. Comm. de la Soc. Math. de Kharkoff 12
(4), 81–87.
TAKHTAJAN, L. (1994). On foundation of the generalized Nambu
mechanics. Commun. Math. Phys. 160, 295–315.
THURSTON, H. A. (1949). Partly associative operations. J. London
Math. Soc. 24, 260–271.
TIMM, J. (1972). Verbandstheoretische Behandlung n-stelliger
Gruppen. Abh. Math. Semin. Univ. Hamb. 37, 218–224.
TRACZYK, T. (1965). Weak isomorphisms of Boolean and Post
algebras. Colloq. Math. 13, 159–164.
TSALLIS, C. (1994). Nonextensive physics: a possible connection
between generalized statistical mechanics and quantum groups.
Phys. Lett. A195, 329–334.
112

Bibliography
TURING, A. (1938). The extensions of a group. Compos. Math. 5,
357–367.
TVERMOES, H. (1953). Über eine Verallgemeinerung des
Gruppenbegriffs. Math. Scand. 1, 18–30.
WANKE-JAKUBOWSKA, M. B. AND M. E. WANKE-JERIE (1984). On
representations of n-groups. Annales Sci. Math. Polonae,
Commentationes Math. 24, 335–341.
ZEKOVI Ć, B. AND V. A. ARTAMONOV (1999). A connection between
some properties of n-group rings and group rings. Math.
Montisnigri 15, 151–158.
ZEKOVI Ć, B. AND V. A. ARTAMONOV (2002). On two problems for
n-group rings. Math. Montisnigri 15, 79–85.
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