Polyadic algebraic structures and their applications

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(inside polyadic products we denote repeated entries by k z ︸︸ ︷ x,...,x as x .... which is another formulation of Fermat's little theorem. Definition 17.4 ( DUPLIJ ...
Polyadic algebraic structures and their applications

S TEVEN D UPLIJ

Munster, Germany ¨

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

Chern Institute of Mathematics - 2018

1

1 History

1 History Ternary algebraic operations (with the arity n

= 3) were introduced by A. Cayley

in 1845 and later by J. J. Sylvester in 1883. ¨ The notion of an n-ary group was introduced in 1928 by D ORNTE [1929] (inspired ¨ by E. Nother). The coset theorem of Post explained the connection between n-ary groups and their covering binary groups P OST [1940]. The next step in study of n-ary groups was the Gluskin-Hosszu´ theorem H OSSZ U´ [1963], G LUSKIN [1965]. The cubic and n-ary generalizations of matrices and determinants were made in K APRANOV

ET AL .

[2003], R AUSCH

[1994], S OKOLOV [1972], physical application in K AWAMURA

DE

T RAUBENBERG [2008].

2

1 History

Particular questions of ternary group representations were considered in B OROWIEC, D UDEK ,

AND

D UPLIJ [2006], D UPLIJ [2012].

Theorems connecting representations of binary and n-ary groups were given in D UDEK

AND

S HAHRYARI [2012].

Ternary fields were developed in D UPLIJ

AND

W ERNER [2015], D UPLIJ [2017b].

In physics, the most applicable structures are the nonassociative Grassmann, ˜ Clifford and Lie algebras L OHMUS

ET AL .

[1994], G EORGI [1999]. The ternary

analog of Clifford algebra was considered in A BRAMOV [1995], and the ternary analog of Grassmann algebra A BRAMOV [1996] was exploited to construct ternary extensions of supersymmetry A BRAMOV

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 AND I ZQUIERDO

[2010].

3

DE

A ZCARRAGA

1 History

The infinite-dimensional version of n-Lie algebras are the Nambu algebras N AMBU [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 F ILIPPOV [1985]. Ternary Filippov algebras were successfully applied to a three-dimensional superconformal gauge theory describing the effective worldvolume theory of coincident M 2-branes of M -theory B AGGER G USTAVSSON [2009].

4

AND

L AMBERT [2008a,b],

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. 9. Polyadic rings and fields 10. Polyadic analogs of the integer number ring Z and the Galois field GF (p). 11. Equal sums of like powers Diophantine equation over polyadic integer numbers

5

3 Notations

3 Notations Let G be a underlying set, universe, carrier, gi

∈ G.

The n-tuple (or polyad) g1 , . . . , gn is denoted by (g1 , . . . , gn ). The Cartesian product G×n consists of all n-tuples (g1 , . . . , gn ). For equal elements g

∈ G, we denote n-tuple (polyad) by (g n ).

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 (n)

The i-projection Pri

The i-diagonal Diagn

n-tuple g 7−→ (g n ).

g

(n)



.

: G×n → G is (g1 , . . . gi , . . . , gn ) 7−→ gi .

: G → G×n sends one element to the equal element

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 (c) G, a “constant” c ∈ G, it is called a 0-ary operation μ0 , which maps the (c) one-point set {•} to G, such that μ0 : {•} → G, and formally has the value (c) μ0 [{•}] = c ∈ G. The 0-ary operation “kills” arity B ERGMAN [1995] μn+m−1 [g, h] = μn [g, μm [h]] .

(3.1)

(c)

Then, if to compose μn with the 0-ary operation μ0 , we obtain (c)

μn−1 [g] = μn [g, c] , 7

(3.2)

3 Notations

because g is a polyad of length (n − 1). It is also seen from the commutative diagram

G

×(n−1)

(c)

× {•}

id×(n−1) ×μ0



(c) μn−1

G×(n−1)

G×n μn

G (c)

which is a definition of a new (n − 1)-ary operation μn−1 .

Remark 3.1. It is important to make a clear distinction between the 0-ary (c)

operation μ0

and its value c in G.

8

(3.3)

4 Polyadic systems

4 Polyadic systems Definition 4.1. A polyadic system G (polyadic algebraic structure) is a set G together with polyadic operations, which is closed under them. Here, we mostly consider concrete polyadic systems with one “chief” (fundamental) n-ary operation μn , which is called polyadic multiplication (or

n-ary multiplication). Definition 4.2. A n-ary system Gn

= hG | μn i 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 hG

system hG arity n0 .

| μn i to construct another polyadic

| μ0n0 i over the same set G, which has multiplication with a different

9

4 Polyadic systems

There are 3 ways to change arity of operation: 1. Iterating. Using composition of the operation μn with itself, one can increase the arity from n to n0iter . Denote the number of iterating multiplications by `

`μ , and use the bold Greek letters μnμ for the resulting composition of n-ary multiplications `μ

μ0n0 = μ`nμ

z }| {     def = μn ◦ μn ◦ . . . μn × id×(n−1) . . . × id×(n−1) ,

(4.1)

where n0

= niter = `μ (n − 1) + 1, which gives the length of a polyad ` ` (g) in the notation μnμ [g]. The operation μnμ is named a long product ¨ D ORNTE [1929] or derived D UDEK [2007].

10

4 Polyadic systems

2. Reducing (Collapsing). Using nc distinguished elements or constants (or nc (ci )

additional 0-ary operations μ0

,i

= 1, . . . nc ), one can decrease arity

from n to n0red (as in (3.2)), such that

(c ...cnc )

μ0n0 = μn01



nc



z }| { def   (c ) (c ) = μn ◦ μ0 1 × . . . × μ0 nc × id×(n−nc )  ,

(4.2)

where

n0 = nred = n − nc ,

(ci )

and the 0-ary operations μ0

(4.3)

can be on any places.

3. Mixing. Changing (increasing or decreasing) arity may be done by combining iterating and reducing (maybe with additional operations of different arity).

11

5 Special elements and properties of n-ary systems

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 P OST [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

h`μ i

=

μ`nμ

h

g

12

`μ (n−1)+1

i

.

(5.2)

5 Special elements and properties of n-ary systems

Example 5.3. Consider a polyadic version of the binary q -addition which appears in study of nonextensive statistics (see, e.g., T SALLIS [1994], N IVANEN

ET AL .

[2003])

μn [g] =

n X

gi + ~

i=1

n Y

gi ,

(5.3)

i=1

∈ C and ~ = 1 − q0 , q0 is a real constant ( q0 6= 1 or ~ 6= 0). It is obvious that g h0i = g , and h i (5.4) g h1i = μn g n−1 , g h0i = ng + ~g n , h i  hk−1i hki n−1 hk−1i n−1 g = μn g ,g = (n − 1) g + 1 + ~g . (5.5) g

where gi

Solving this recurrence formula for we get

g

hki



n − 1 1−n =g 1+ g ~



1 + ~g

13

 n−1 k

n − 1 2−n − . g ~

(5.6)

5 Special elements and properties of n-ary systems

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 h`μ i = 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`nμ [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

5 Special elements and properties of n-ary systems

Definition 5.6. A polyadic (n-ary) identity (or neutral element) of a polyadic (ε)

system is a distinguished element ε (and the corresponding 0-ary operation μ0 ) such that for any element g

∈ G we have ROBINSON [1958]  n−1  μn g, ε = 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, 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

(5.10)

5 Special elements and properties of n-ary systems

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 h`μ i = 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

μ2n [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 T HURSTON [1949] and studied in B ELOUSOV [1972], S OKHATSKY [1997]. Definition 5.8. A polyadic semigroup (n-ary semigroup) is a n-ary system in which the operation is associative, or Gsemigrp n 16

= hG | μn | associativityi.

5 Special elements and properties of n-ary systems

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 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

(5.14)

5 Special elements and properties of n-ary systems

It is very important to find the associativity preserving conditions, where an associative initial operation μn leads to an associative final operation μ0n0 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 (c)

μ2 [g, h] = μn [g, c, h] .

(5.15)

Associativity preserving mixing constructions with different arities and places were considered in D UDEK

AND

M ICHALSKI [1984], M ICHALSKI [1981],

S OKHATSKY [1997]. Definition 5.11. A totally associative polyadic system with identity ε, satisfying (5.9) μn



g, ε

n−1



= g is called a polyadic monoid.

The structure of any polyadic monoid is fixed P OP AND P OP [2004]: iterating a ˇ UPONA AND T RPENOVSKI [1961]. binary operation C Several analogs of binary commutativity of polyadic system. 18

5 Special elements and properties of n-ary systems

A polyadic system is σ -commutative, if μn

= μn ◦ σ

(5.16) μn [g] = μn [σ ◦ g] ,  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, h

n−1





= μn h

n−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

5 Special elements and properties of n-ary systems

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 ( E VANS [1963], B ELOUSOV [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 G ŁAZEK AND G LEICHGEWICHT [1982].

20

5 Special elements and properties of n-ary systems

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

5 Special elements and properties of n-ary systems

In a polyadic group the only solution of (5.22) μn [u, h, t]

= g is called a

¨ [1929] querelement of g and denoted by g ˉ D ORNTE

μn [h, gˉ] = g,

(5.23)

ˉ can be on any place. Any idempotent g coincides with its querelement where g gˉ = g . It follows from (5.23) and (5.10), that the polyad  n−2 ng = g gˉ

(5.24)

is neutral for any element of a polyadic group, where g ˉ can be on any place. The

number of relations in (5.23) can be reduced from n (the number of possible ¨ places) to only 2 (when g is on the first and last places D ORNTE [1929], T IMM ¨ [1972], or on some other 2 places ). In a polyadic group the Dornte relations

μn [g, nh;i ] = μn [nh;j , g] = g hold true for any allowable i, j . Analog of g

22

∙ h ∙ h−1 = h ∙ h−1 ∙ g = g .

(5.25)

5 Special elements and properties of n-ary systems

The relation (5.23) can be treated as a definition of the unary queroperation

μ ˉ 1 [g] = gˉ.

(5.26)

Definition 5.17. A polyadic group is a universal algebra grp

Gn

¨ relationsi , = hG | μn , μ ˉ 1 | associativity, Dornte

(5.27)

where μn is n-ary associative operation and μ ˉ 1 is the queroperation (5.26), such that the following diagram

G

×(n)

id ×Diag(n−1)

G×G

μ1 id×(n−1) ×ˉ

G

×n

μ ˉ 1 ×id×(n−1)

μn Pr1

G

Pr2

G×n Diag(n−1) ×id

G×G

(5.28)

commutes, where μ ˉ 1 can be only on the first and second places from the right (resp. left) on the left (resp. right) part of the diagram.

23

5 Special elements and properties of n-ary systems

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 μnμ (4.1) (cf. D UDEK [1980] for `μ = 2 and G AL’ MAK [2007]). Definition 5.18. Let us define the querpower k of g recursively hhkii

gˉ where g ˉhh0ii

=

gˉhhk−1ii



,

(5.29)

= g , gˉhh1ii = gˉ, or as the k composition k

μ ˉ ◦k 1

z }| { ˉ1 ◦ μ =μ ˉ1 ◦ . . . ◦ μ ˉ 1 of the queroperation (5.26).

For instance, μ ˉ ◦2 1 has g ˉ

, such that for any ternary group μ = μn−3 ˉ ◦2 n 1 = id, i.e. one

= g.

24

5 Special elements and properties of n-ary systems

The negative polyadic power of an element g by (after use of (5.2))

h

μn g

h`μ −1i

,g

n−2

,g

h−`μ i

i

μ`nμ

= g,

h

g

`μ (n−1)

,g

h−`μ i

i

= g. (5.30)

Connection of the querpower and the polyadic power by the Heine numbers H EINE [1878] or q -numbers K AC

AND

C HEUNG [2002]

qk − 1 [[k]]q = , q−1 which have the “nondeformed” limit q

(5.31)

→ 1 as [k]q → k . Then

gˉhhkii = g h−[[k]]2−n i ,

(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

6 (One-place) homomorphisms of polyadic systems

= hG | μn i and G0n0 = hG0 | μ0n0 i be two polyadic systems of any kind

Let Gn

(quasigroup, semigroup, group, etc.). If they have the multiplications of the same arity n

= n0 , then one can define the mappings from Gn to G0n . 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 An ordered system of mappings 0 ϕGG n+1

(μn [g1 , . . . , gn ]) =

n

μ0n

0

ϕGG i h

o

0 ϕGG 1

26

0

: G → G0 , i = 1, . . . , n + 1.

is called a homotopy from Gn to G0n , if 0 (g1 ) , . . . , ϕGG n

i

(gn ) ,

gi ∈ G.

(6.1)

6 (One-place) homomorphisms of polyadic systems

In general, one should add to this definition the “mapping” of the multiplications (μμ0 )

μn

ψ nn0

7→ μ0n0 .

(6.2)

In such a way,  the homotopycan be defined as the (extended) system of mappings

(μμ0 ) GG0 ϕi ; ψ nn

. The corresponding commutative (equiary) diagram

is

0

ϕGG n+1

G μn

G0

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

G×n

0 0 ϕGG ×...×ϕGG 1 n

μ0n

(G0 )

(6.3)

×n

0

If all the components ϕGG of a homotopy are bijections, it is called an isotopy. In i 0

case of polyadic quasigroups B ELOUSOV [1972] all mappings ϕGG are usually i taken as permutations of the same underlying set G If the multiplications are also coincide μn

=

μ0n , then the set

called an autotopy of the polyadic system Gn .

27

= G0 .



ϕGG i ; id



is

6 (One-place) homomorphisms of polyadic systems

The diagonal counterparts of homotopy, isotopy and autotopy (when all mappings

ϕGG coincide) are homomorphism, isomorphism and automorphism. i A homomorphism from Gn to G0n is given, if there exists one mapping 0 ϕGG : G → G0 satisfying

h

0

0

0

i

ϕGG (μn [g1 , . . . , gn ]) = μ0n ϕGG (g1 ) , . . . , ϕGG (gn ) , Usually the homomorphism is denoted by thesame one letter ϕ extended pair of mappings

ϕ

GG0

(μμ0 )

; ψ nn

GG0

gi ∈ G.

(6.4)

or the

.

They “...are so well known that we shall not bother to define them carefully” H OBBY

AND

M C K ENZIE [1988].

28

7 Standard Hosszu-Gluskin theorem ´

7 Standard Hosszu-Gluskin ´ theorem Consider concrete forms of polyadic multiplication in terms of lesser arity operations. History. Simplest way of constructing a n-ary product μ0n from the binary one

μ2 = (∗) is `μ = n iteration (4.1) S USCHKEWITSCH [1935], M ILLER [1935] μ0n [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 = hG | μ2 = ∗, ei was called a covering group of the n-ary group G0n = hG | μ0n i in P OST [1940] (also, T VERMOES [1953]), where a theorem establishing a more general (than (7.1)) structure of μ0n [g] in terms of subgroup structure was given.

29

7 Standard Hosszu-Gluskin theorem ´

A manifest form of the n-ary group product μ0n [g] in terms of the binary one and a special mapping was found in H OSSZ U´ [1963], G LUSKIN [1965] and is called the Hosszu-Gluskin theorem, despite the same formulas having appeared much ´ earlier in T URING [1938], P OST [1940] (relationship between all the formulations in G AL’ MAK

AND

VOROBIEV [2013]).

Rewrite (7.1) in its equivalent form

μ0n [g] = g1 ∗ g2 ∗ . . . ∗ gn ∗ e,

gi , e ∈ G,

where e is a distinguished element of the binary group hG

(7.2)

| ∗, ei, that is the

identity. Now we apply to (7.2) an “extended” version of the homotopy relation

(6.1) with Φi

= ψ i , i = 1, . . . n, and the l.h.s. mapping Φn+1 = id, but add an action ψ n+1 on the identity e μn [g] = μ(e) n [g] = ψ 1 (g1 ) ∗ ψ 2 (g2 ) ∗ . . . ∗ ψ n (gn ) ∗ ψ n+1 (e) . (7.3)

30

7 Standard Hosszu-Gluskin theorem ´

The most general form of polyadic multiplication in terms of (n + 1) “extended” homotopy maps ψ i , i

G

×(n)

= 1, . . . n + 1, the diagram (e)

× {•}

id×n ×μ0

G×(n+1)

ψ 1 ×...×ψ n+1



G×(n)

(e) μn

G×(n+1) μ×n 2

(7.4)

G

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

7 Standard Hosszu-Gluskin theorem ´

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 ψ i in terms of μn itself S OKOLOV [1976]. A simplest binary multiplication (c)

which can be built from μn is (recall (5.15) μ2

[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 hG

| μn i, so formally t = e−1 .

32

7 Standard Hosszu-Gluskin theorem ´

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

hG | μn i 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., B ERGMAN [2012]), which can be called a polyadic Maltsev term and is defined as



def

p (g, e, h) = μn g, e

−1

having the standard term properties p (g, e, e) For n-ary group we can write g−1

g

−1

is g

−1



= μn e, g

S HCHUCHKIN [2003]

n−3



,h



(7.10)

= g, p (e, e, h) = h.  n−3 = g , gˉ and the binary group inverse

, gˉ, e , the polyadic Maltsev term becomes 

p (g, e, h) = μn g, e 33

n−3



, eˉ, h .

(7.11)

7 Standard Hosszu-Gluskin theorem ´

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] = ψ 1 (g) ∗ ψ 2 (h) ∗ ψ 3 (u) ∗ ψ 4 (e)

(7.12)

and try to construct ψ i in terms of the ternary product μ3 and the binary identity k

k



e. A neutral ternary polyad (ˉ e, e) or its powers eˉ , e . Thus, taking for all insertions the minimal number of neutral polyads, we get





∗ ∗ ∗     ↓ ↓  ↓    μ3 [g, h, u] = μ73 g, eˉ , e, h, eˉ, eˉ , e, e, u, eˉ, eˉ, eˉ , e, e, e .      

34

(7.13)

7 Standard Hosszu-Gluskin theorem ´

We rewrite (7.13) as





∗ ∗ ∗     ↓ ↓  ↓   3 2 μ3 [g, h, u] = μ3 g, eˉ , μ3 [e, h, eˉ] , eˉ , μ3 [e, e, u, eˉ, eˉ] , eˉ , μ3 [e, e, e] .       (7.14)

Comparing this with (7.12), we can identify

ψ 1 (g) = g,

(7.15)

ψ 2 (g) = ϕ (g) ,

(7.16)

ψ 3 (g) = ϕ (ϕ (g)) = ϕ2 (g) ,

(7.17)

ψ 4 (e) = μ3 [e, e, e] = eh1i ,

(7.18)

ϕ (g) = μ3 [e, g, eˉ] .

35

(7.19)

7 Standard Hosszu-Gluskin theorem ´

Thus, we get the Hosszu-Gluskin “chain formula” for n ´

=3

μ3 [g, h, u] = g ∗ ϕ (h) ∗ ϕ2 (u) ∗ b, b = eh1i .

(7.21)

is a fixed point, because ϕ

e

h1i



= eh1i , as well as  2k+1  hki k = μ3 e higher polyadic powers e of the binary identity e are obviously  also fixed points ϕ ehki = ehki .

The polyadic power e

h1i

(7.20)

By analogy, the Hosszu-Gluskin “chain formula” for arbitrary n can be obtained ´ using substitution e ˉ→



e

 −1 k



e

−1

, neutral polyads

e

−1

, e and their powers

, ek , the mapping ϕ in the n-ary case is 

ϕ (g) = μn e, g, e



−1



,

and μn [e, . . . , e] is also the first n-ary power eh1i (5.2).

36

(7.22)

7 Standard Hosszu-Gluskin theorem ´

In this way, we obtain the Hosszu-Gluskin “chain formula” for arbitrary n ´

μn [g1 , . . . , gn ] = g1 ∗ϕ (g2 )∗ϕ2 (g3 )∗. . .∗ϕn−2 (gn−1 )∗ϕn−1 (gn )∗eh1i .

(7.23)

Thus, we have found the “extended” homotopy maps ψ i from (7.3)

ψ i (g) = ϕi−1 (g) , i = 1, . . . , n, ψ n+1 (g) = g h1i , where by definition ϕ0 (g)

(7.24) (7.25)

= g . Using (7) and (7.23) we can formulate the

standard Hosszu-Gluskin theorem in the language of polyadic powers. ´

= hG | μn , μ ˉ 1 i one can define a binary group G∗2 = hG | μ2 = ∗, ei and its automorphism ϕ such that the

Theorem 7.2. On a polyadic group Gn

Hosszu-Gluskin “chain formula” ( 7.23) is valid. ´

37

7 Standard Hosszu-Gluskin theorem ´

The following reverse Hosszu-Gluskin theorem holds. ´ Theorem 7.3. If in a binary group G∗2 automorphism ϕ such that

= hG | μ2 = ∗, ei one can define an

ϕn−1 (g) = b ∗ g ∗ b−1 , ϕ (b) = b,

where b

(7.26) (7.27)

∈ 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

= eh1i .

38

8 “Deformation” of Hosszu-Gluskin chain formula ´

8 “Deformation” of Hosszu-Gluskin ´ chain formula Idea: to generalize the Hosszu-Gluskin chain formula D UPLIJ [2016]. We take the ´ number of the inserted neutral polyads arbitrarily, not only minimally, as they are all neutral. Indeed, in the particular case n

= 3, we put the map ϕ as

` (q)

ϕq (g) = μ3ϕ

[e, g, eˉq ] ,

(8.1)

where the number of multiplications

`ϕ (q) =

q+1 2

= 1, 2, 3 . . ., while q = 1, 3, 5, 7 . . .. Then, we get h i • q q q+1 q(q+1)+1 . μ3 [g, h, u] = μ3 g, eˉ, (e, h, eˉ ) , eˉ, (e, u, eˉ ) , eˉ, e

(8.2)

is an integer `ϕ (q)

39

(8.3)

8 “Deformation” of Hosszu-Gluskin chain formula ´

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

ψ 1 (g) =

[[0]]q ϕq

(g) = ϕ0q (g) = g,

ψ 2 (g) = ϕq (g) = ψ 3 (g) =

ϕq+1 q

[[1]]q ϕq

(g) =

(8.4)

(g) ,

(8.5)

[[2]]q ϕq

(g) , h i h i ψ 4 (g) = μ•3 g q(q+1)+1 = μ•3 g [[3]]q ,

(8.6) (8.7)

where ϕ is defined by (8.1) and [[k]]q is the q -deformed number and we put

ϕ0q = id. The corresponding “q -deformed” chain formula (for n = 3) can be written as

μ3 [g, h, u] = g ∗

[[1]]q ϕq

bq = eh`e (q)i ,

`e (q) = q

[[2]]q 2

.

40

(h) ∗

[[2]]q ϕq

(u) ∗ bq ,

(8.8) (8.9) (8.10)

8 “Deformation” of Hosszu-Gluskin chain formula ´

chain → 1 of (8.8) gives the standard Hosszu-Gluskin ´ formula (7.20) for n = 3. For arbitrary n we insert all possible powers of neutral   −1 k polyads , ek (they are allowed by the chain properties), and obtain e The “nondeformed” limit q

ϕq (g) =

μn`ϕ (q)

h

e, g, e

 −1 q

i

,

(8.11)

q (n − 2) + 1 is an integer and where the number of multiplications `ϕ (q) = n−1 `ϕ (q) → q , as n → ∞, and `ϕ (1) = 1, as in (7.22).

41

8 “Deformation” of Hosszu-Gluskin chain formula ´

The “deformed” map ϕq is a kind of a-quasi-endomorphism G LUSKIN

AND

S HVARTS [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) , where a

(8.12)

= ϕq (e). A general quasi-endomorphism D UPLIJ [2016]  ϕq (g) ∗ ϕq (h) = ϕq g ∗ ϕq (e) ∗ h .

(8.13)

The corresponding diagram

G×G ϕq ×ϕq

G×G

μ2



ϕq

G G × {•} × G

(e) id ×μ0 ×id

commutes. If q

G μ2 ×μ2

(8.14)

G×G×G

= 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

8 “Deformation” of Hosszu-Gluskin chain formula ´

The “extended” homotopy maps ψ i (7.3) now are [[0]]q

ψ 1 (g) = ϕq

(g) = ϕ0q (g) = g, [[1]]q

ψ 2 (g) = ϕq (g) = ϕq ψ 3 (g) =

ϕq+1 q

(g) =

(8.15)

(g) ,

[[2]]q ϕq

(8.16)

(g) ,

(8.17)

.. .

ψ n−1 (g) = ϕqq ψ n (g) =

n−3

+...+q+1

[[n−2]]q

(g) = ϕq

q n−2 +...+q+1 ϕq

(g) ,

[[n−1]]q ϕq

(g) = (g) , h n−1 i h i [[n]]q • q +...+q+1 • = μn g . ψ n+1 (g) = μn g

(8.18) (8.19) (8.20)

In terms of the polyadic power (5.2), the last map is

ψ n+1 (g) = g

h`e i

,

`e (q) = q

43

[[n − 1]]q n−1

.

(8.21)

8 “Deformation” of Hosszu-Gluskin chain formula ´

Thus the “q -deformed” n-ary chain formula is D UPLIJ [2016]

μn [g1 , . . . , gn ] = g1 ∗ ∗

[[1]]q ϕq

[[n−2]]q ϕq

In the “nondeformed” limit q chain formula (7.23).

(g2 ) ∗

(gn−1 ) ∗

[[2]]q ϕq

(g3 ) ∗ . . .

[[n−1]]q ϕq

(gn ) ∗ eh`e (q)i .

(8.22)

→ 1 (8.22) reproduces the standard Hosszu-Gluskin ´

Instead of the fixed point relation (7.27) ϕ (b)

= b we now have the quasi-fixed

point

ϕq (bq ) = bq ∗ ϕq (e) , h i bq = μ•n e[[n]]q = eh`e (q)i .

The conjugation relation (7.26) ϕn−1 (g)

(8.23) (8.24)

= b ∗ g ∗ b−1 in the “deformed” case

becomes the quasi-conjugation D UPLIJ [2016] [[n−1]]q

ϕq

[[n−1]]q

(g) ∗ bq = bq ∗ ϕq 44

(e) ∗ g.

(8.25)

8 “Deformation” of Hosszu-Gluskin chain formula ´

We formulate the following “q -deformed” analog of the Hosszu-Gluskin theorem ´ D UPLIJ [2016].

= hG | μn , μ ˉ 1 i one can define a binary group G∗2 = hG | μ2 = ∗, ei and (the infinite “q -series” of) its automorphism ϕq Theorem 8.1. On a polyadic group Gn

such that the “deformed” chain formula ( 8.22) is valid

μn [g1 , . . . , gn ] =



n Y

ϕ[[i−1]]q (gi )

i=1

!

∗ 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 [[n−1]]q ϕq

(g) = bq ∗

[[n−1]]q ϕq

45

(e) ∗ g ∗ b−1 q .

(8.27)

9 (One-place) generalizations of homomorphisms

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

z

n

}| { ϕ ϕ1 0 ϕ2 n−1 00 ϕn 000 Gn → Gn → . . . → Gn → Gn

(9.1)

Generalized P OST [1940] n-adic substitutions in G AL’ 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

9 (One-place) generalizations of homomorphisms

The first way leads to the concept of weak homomorphism which was introduced in G OETZ [1966], M ARCZEWSKI [1966], G ŁAZEK

AND

M ICHALSKI [1974] for

non-indexed algebras and in G ŁAZEK [1980] for indexed algebras, then developed in T RACZYK [1965] for Boolean and Post algebras, in D ENECKE [2009] for coalgebras and F -algebras D ENECKE

AND

AND

W ISMATH

S AENGSURA [2008].

| μn i and hG0 | μ0n0 i the following ×n0 0 additional term operations of opposite arity ν n : G → G and ν 0n : G0×n → G0 and consider two equiary mappings between hG | μn , ν n0 i and hG0 | μ0n0 , ν 0n i. Incorporate into the polyadic systems hG

47

9 (One-place) generalizations of homomorphisms

| μn , ν n0 i to hG0 | μ0n0 , ν 0n i is given, if there 0 exists a mapping ϕGG : G → G0 satisfying two relations simultaneously h i 0 0 0 (9.2) ϕGG (μn [g1 , . . . , gn ]) = ν 0n ϕGG (g1 ) , . . . , ϕGG (gn ) , h i 0 0 0 ϕGG (ν n0 [g1 , . . . , gn0 ]) = μ0n0 ϕGG (g1 ) , . . . , ϕGG (gn0 ) . (9.3) A weak homomorphism from hG

ϕGG

G μn

G

0

G0

(μν 0 ) ............ .......... ψ nn ×n



ϕ

GG0

×n

ϕGG

G

ν 0n

ν n0

0 ×n

(G )

G

0

G0

0

) ............. ............ ψ (νμ 0 0 ×n0



n n ×n0 GG0

ϕ

μ0n0

(9.4)

0 0 ×n

(G )

If only one of the relations (9.2) or (9.3) holds, such a mapping is called a 0

semi-weak homomorphism KOLIBIAR [1984]. If ϕGG is bijective, then it defines a weak isomorphism.

48

10 Multiplace mappings and heteromorphisms

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 hG

| μn i as

id ) μ(` = μn × (idG ) n

×`id

,

(10.1)

×(n+`id ) ×(1+`id ) id ) μ(` : G → G . n (`id )

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

(n,n0 )

Introduce a multiplace mapping Φk

(n,n0 )

Φk

(10.2)

(i).

acting as D UPLIJ [2012]

: G×k → G0 .

49

(10.3)

10 Multiplace mappings and heteromorphisms

We have the following commutative diagram which changes arity

G

×k

Φk

(` ) μn id

G

×n0

×kn (Φk )

G0 μ0n0

(10.4)

0 0 ×n

(G )

Definition 10.1. A k -place heteromorphism from Gn to G0n0 is given, if there

(n,n0 )

exists a k -place mapping Φk

(10.3) such that the corresponding defining

equation (a modification of (6.4)) depends on the place i of μn in (10.1).

50

10 Multiplace mappings and heteromorphisms

For i

= 1 it can read as D UPLIJ [2012] 

  (n,n0 )  Φk   

μn [g1 , . . . , gn ] gn+1 .. .

gn+`id







  n,n0   )  0  ( Φ = μ   n0  k    

g1





  (n,n0 )  ..   .  , . . . , Φk   gk

gk(n0 −1)

(10.5)

= 3, n0 = 2, k = 2, `id = 1 we have        (3,2)  μ3 [g1 , g2 , g3 ]  (3,2)  g3  0  (3,2)  g1  Φ2 = μ 2 Φ2 , Φ2 . g4 g2 g4

In the particular case n

(10.6)

This was used in the construction of the bi-element representations of ternary groups B OROWIEC, D UDEK ,

AND

D UPLIJ [2006], D UPLIJ [2012].

51

.. .

gkn0



   . 

10 Multiplace mappings and heteromorphisms

= M2adiag (K), a set of antidiagonal 2 × 2 matrices over 0 and the field K G = K, where K = R, C, Q, H. For the elements 

Example 10.2. Let G

gi = 

0

ai

bi

0

, i = 1, 2, we construct a 2-place mapping G × G → G0 as (3,2) Φ2

 

g1 g2



 = a 1 a 2 b 1 b2 ,

(10.7)

which satisfies ( 10.6). Introduce a 1-place mapping by ϕ (gi )

= ai bi , which satisfies the standard ( 6.4) for a commutative field K only (= R, C) becoming a (3,2) and ϕ homomorphism. The relation between the heteromorhism Φ2   (3,2)  g1  Φ2 = ϕ (g1 ) ∙ ϕ (g2 ) = a1 b1 a2 b2 , (10.8) g2 where the product (∙) is in K, such that ( 6.4) and ( 10.6) coincide. For the

noncommutative field K (=

Q or H) we can define only the heteromorphism. 52

10 Multiplace mappings and heteromorphisms

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

(n,n0 )

multiplications in the argument of Φk

in its defining relation. We define a

`μ -ple `id -intact id-product for hG; μn i as μ ,`id ) μ(` = (μn ) n

×`μ

× (idG )

×`id

,

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

(10.9) (10.10)

A `μ -ple k -place heteromorphism from Gn to G0n0 is given, if there exists a

(n,n0 )

k -place mapping Φk

(10.3).

53

10 Multiplace mappings and heteromorphisms

The main heteromorphism equation is D UPLIJ [2012] 

         0 n,n   Φ  k        

μn [g1 , . . . , gn ] ,



. . .

μn g

n `μ −1

, . . . , g

gn` +1 , μ . . .

gn` +` μ id

            

n`μ

`id



       



   `μ                0   n,n   = μ0 0  Φ  n   k         

      

g1 . . .

gk

     n,n0      , . . . , Φk     



g

k n0 −1 . . .

g

kn0

(10.11)

∗ g2 ) = Φ (g1 ) • Φ (g2 ), which corresponds to n = 2, n0 = 2, k = 1, `μ = 1, `id = 0, μ2 = ∗, μ02 = •. It is a polyadic analog of binary Φ (g1

We obtain two arity changing formulas

n−1 `id , k n−1 n0 = `μ + 1, k n0 = n −

where n−1 k `id

≥ 1 and

n−1 k `μ

≥ 1 are integer. 54

(10.12) (10.13)





    .  

10 Multiplace mappings and heteromorphisms

The following inequalities hold valid

1 ≤ `μ ≤ k,

0 ≤ `id ≤ k − 1,

`μ ≤ k ≤ (n − 1) `μ , 2 ≤ n0 ≤ n.

(10.14) (10.15) (10.16) (10.17)

The main statement follows from (10.17):

(n,n0 )

The heteromorphism Φk If `id If `id

decreases arity of the multiplication.

6= 0 then it is change of the arity n0 6= n.

= 0, then k = kmin = `μ , and no change of arity n0max = 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 n0min = 2 (the final operation is binary), corresponds to the maximal value of places k = kmax = (n − 1) `μ . 55

10 Multiplace mappings and heteromorphisms

Figure 1:

n0 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)).

Dependence of the final arity

56

10 Multiplace mappings and heteromorphisms

Theorem 10.3. Any n-ary system can be mapped into a binary system by (n,2)

binarizing heteromorphism Φ(n−1)` , `id μ

= (n − 2) `μ .

Proposition 10.4. Classification of `μ -ple heteromorphisms: 1.

2.

(n,n)

n0 = n0max = n =⇒ Φ`μ k = kmin = `μ . 0

(n,n0 )

2 < n < n =⇒ Φk

is the `μ -place homomorphism,

is the intermediate heteromorphism with

n−1 k= 0 `μ , n −1 3.

(n,2)

n − n0 `id = 0 `μ . n −1

n0 = n0min = 2 =⇒ Φ(n−1)`μ is the (n − 1) `μ -place binarizing heteromorphism, i.e., k = kmax = (n − 1) `μ .

57

(10.18)

10 Multiplace mappings and heteromorphisms

Table 1: “Quantization” of heteromorphisms

k 2 3 3

`μ 1 1 2

1 2 1

4

1

3

4

2

2

4

3

n/n0

`id

1

n=

3,

5,

7,

...

n0 =

2,

3,

4,

...

n=

4,

7,

10,

...

n0 =

2,

3,

4,

...

n=

4,

7,

10,

...

n0 =

3,

5,

7,

...

n=

5,

9,

13,

...

n0 =

2,

3,

4,

...

n=

3,

5,

7,

...

n0 =

2,

3,

4,

...

n=

5,

9,

13,

...

n0 =58 4,

7,

10,

...

11 Associativity, quivers and heteromorphisms

11 Associativity, quivers and heteromorphisms Semigroup heteromorphisms: associativity of the final operation μ0n0 , 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 D UPLIJ [2012]. For instance, for the multiplication μ4 [g1 , h2 , g2 , u1 ] the 4-adic quiver is denoted by

{g1 → h2 → g2 → u1 }. Define polyadic quivers which are related to the main heteromorphism equation (10.11).

59

11 Associativity, quivers and heteromorphisms

For example, the polyadic quiver {g1

→ h2 → g2 → u1 ; h1 , u2 } corresponds to the heteromorphism with n = 4, n0 = 2, k = 3, `id = 2 and `μ = 1 such that

(4,2) Φ3

   

μ4 [g1 , h2 , g2 , u1 ] h1 u2







g1







g2

       = μ02 Φ(4,2)  h1  , Φ(4,2)  h2  .   3   3   u1 u2 (11.1)

As it is seen from here (11.1), the product μ02 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 μ0n0 in the main heteromorphism equation (10.11), when the initial multiplication μn is associative.

60

11 Associativity, quivers and heteromorphisms

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

(11.2)

corr



(4,2)   Φ3 

μ4 [g1 , h2 , u1 , g2 ] h1 u2













g1 g2        = μ0 Φ(4,2)  h1  , Φ(4,2)  h2  . 2 3 3      u1 u2

We propose a classification of associative polyadic quivers and the rules of construction of corresponding heteromorphism equations, i.e. consistent procedure for building semigroup heteromorphisms.

61

11 Associativity, quivers and heteromorphisms

The first class of heteromorphisms (`id

= 0 or intactless), that is `μ -place (multiplace) homomorphisms. As an example, for n = n0 = 3, k = 2, `μ = 2

we have (3,3) Φ2

 

μ3 [g1 , g2 , g3 ] μ3 [h1 , h2 , h3 ]







  = μ03 Φ(3,3) 2

g1 h1





 , Φ(3,3)  2

g2 h2





 , Φ2(3,3) 

g3 h3

(11.3)





Note that the analogous quiver with opposite arrow directions is (3,3) Φ2

 

μ3 [g1 , g2 , g3 ] μ3 [h3 , h2 , h1 ]







  = μ03 Φ(3,3) 2

g1 h1





 , Φ(3,3)  2

g2 h2

It was used in constructing the middle representations of ternary groups B OROWIEC, D UDEK ,

AND

D UPLIJ [2006], D UPLIJ [2012].

62





 , Φ2(3,3)  (11.4)

g3 h3





11 Associativity, quivers and heteromorphisms

An important class of intactless heteromorphisms (with `id

= 0) preserving

associativity can be constructed using an analogy with the Post substitutions P OST [1940], and therefore we call it the Post-like associative quiver. The number of places k is now fixed by k

= n − 1, while n0 = 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 (3,3) Φ2

 

μ3 [g1 , h2 , g3 ] μ3 [h1 , g2 , h3 ]







 = μ03 Φ(3,3)  2

g1 h1





 , Φ(3,3)  2

g2 h2



 , Φ(3,3)  2 (11.5)

This construction appeared in the study of ternary semigroups of morphisms C HRONOWSKI [1994]. Its n-ary generalization was used special representations of n-groups G LEICHGEWICHT

ET AL .

[1983], WANKE -J AKUBOWSKA

AND

WANKE -J ERIE [1984] (where the n-group with the multiplication μ02 was called the diagonal n-group).

63



g3 h3





12 Multiplace representations of polyadic systems

12 Multiplace representations of polyadic systems In the heteromorphism equation final multiplication μ0n0 is a linear map, which leads to restrictions on the final polyadic structure G0n0 .

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

= hG; μ2 i 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) , where g, h

∈ G and (∗) is the binary multiplication in End V .

64

(12.1)

12 Multiplace representations of polyadic systems

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.

n0 6= n.

Consider the structure of the final n0 -ary multiplication μ0n0 in (10.11), taking into

account that the final polyadic system G0n0 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 G0n0 = dern0 (End V ), as it was proposed for the ternary case in B OROWIEC, D UDEK , AND D UPLIJ [2006], D UPLIJ [2012].

65

12 Multiplace representations of polyadic systems

In this way G0n0 becomes a derived n0 -ary (semi)group of endomorphisms of V with the multiplication μ0n0

: (End V )

×n0

→ End V , where

μ0n0 [v1 , . . . , vn0 ] = v1 ∗ . . . ∗ vn0 , vi ∈ End V.

(12.3)

Because the multiplication μ0n0 (12.3) is derived and is therefore associative, consider the associative initial polyadic systems.

= hG | μn i be an associative n-ary polyadic system. By analogy with (10.3), we introduce the following k -place mapping Let Gn

(n,n0 )

Πk

: G×k → End V.

66

(12.4)

12 Multiplace representations of polyadic systems

A multiplace representation of an associative polyadic system Gn in a vector

(n,n0 )

which satisfies space V is given, if there exists a k -place mapping (12.4) Πk the (associativity preserving) heteromorphism equation (10.11), that is D UPLIJ [2012] 

         0 n,n   Π  k        

μn [g1 , . . . , gn ] ,



μn g

. . .

n `μ −1

, . . . , g

gn` +1 , μ . . .

gn` +` μ id

            

n`μ



`id

       



   z `μ               n,n0   = Π  k        

G×k

Πk

(`μ ,`id ) μn

G (`μ ,`id )

where μn

×kn0

×n0

(Πk )

      

g1 . . .

gk

n0 }|



   0  n,n   ∗ . . . ∗ Πk  

     

g

k n0 −1 . . .

g

kn0

(12.5)

End V (∗)n

(End V )

0

×n0

is `μ -ple `id -intact id-product given by (10.9).

67



(12.6)



{ 

   ,  

12 Multiplace representations of polyadic systems

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 (min)

with n0

= n0max = n, without intact elements lid = lid minimal number of places k = kmin = `μ .

= 0, and

2. The intact element multiplace representation which is the intermediate heteromorphism with 2

< n0 < n and the number of intact elements is n − n0 lid = 0 `μ . n −1

(12.7)

3. The binary multiplace representation which is a binarizing heteromorphism (3) with n0 (max)

lid

= n0min = 2, the maximal number of intact elements = (n − 2) `μ and maximal number of places k = kmax = (n − 1) `μ .

(12.8)

In case of n-ary groups, we need an analog of the “normalizing” relation (12.2)

68

12 Multiplace representations of polyadic systems

Π1 (e) = idV . If the n-ary group has the unity e, then    e       0 (n,n )  ..  k  = idV . Πk  .      e 

69

(12.9)

12 Multiplace representations of polyadic systems

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

Π1 h as follows

for all h



     0  (n,n )  Πk      

−1



∗ h = idV ,

 ˉ  h    .. .

   ˉ  h  h     .. .

   h 



`id

(12.10)



       = idV ,      

(12.11)

ˉ is the querelement of h which can be on any places in the ∈Gn , where h

¨ l.h.s. of (12.11) due to the Dornte identities.

70

12 Multiplace representations of polyadic systems

¨ 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



g1  (n,n0 )  .. Πk  .  gk



       0  (n,n )    = Πk       



t1 .. .

t`μ

 g     .. .

   g 

`id

where g is an arbitrary fixed element of the n-ary group and

      ,      

ta = μn [ga1 , . . . , gan−1 , gˉ] , a = 1, . . . , `μ .

71

(12.12)

(12.13)

13 Multiactions and G-spaces

13 Multiactions and G-spaces

= hG | μn i be a polyadic system and X be a set. A (left) 1-place action (n) of Gn on X is the external binary operation ρ1 : 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, n n o o (n) (n) (n) (n) ρ1 {μn [g1 , . . . gn ] |x} = ρ1 g1 |ρ1 g2 | . . . |ρ1 {gn |x} . . . . Let Gn

(13.1)

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

∈ G (which may or may not coincide with the unity (n) (n) of Gn ) that ρ1 {ex |x} = x for all x ∈ X, and the mapping x 7→ ρ1 {ex |x} is a bijection of X. The right 1-place actions of Gn on X are defined in a implied the there exist such ex

symmetric way, and therefore we will consider below only one of them.

72

13 Multiactions and G-spaces

(n) Obviously, we cannot compose ρ1 and

(n0 ) ρ1 with

n 6= n0 .

Usually X is called a G-set or G-space depending on its properties (see, e.g., ¨ H USEM 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 polyadic operation

= hG | μn i and a set X we introduce an external ρk : G×k × X → X,

(13.2)

which is called a (left) k -place action or multiaction. We use the analogy with multiplication laws of the heteromorphisms (10.11) . and the multiplace representations (12.5).

73

13 Multiactions and G-spaces

We propose (schematically) D UPLIJ [2012]

ρ

                   

(n) k                   

μn [g1 , . . . , gn ] ,



μn g

. . .

n `μ −1

, . . . , g

gn` +1 , μ . . .

gn` +` μ id

            

n`μ

`id



       

         z  `μ                         (n) x = ρ  k                        

g1 . . .

gk



...

(n) ρ k

n0 }|             

g

k n0 −1 . . .

g



kn0

{               x ... .             (13.3)

The connection between all the parameters here is the same as in the arity changing formulas (10.12)–(10.13). Composition of mappings is associative, and therefore in concrete cases we can use our associative quiver technique from Section 11.

74

13 Multiactions and G-spaces

If Gn is n-ary group, then we should add to (13.3) the “normalizing” relations. If there is a unity e ∈Gn , then

    e       . (n) .. x = x, ρk       e  

for all x

∈ X.

(13.4)

In terms of the querelement, the normalization has the form

(n)

ρk

                          

 ˉ  h    .. .

   ˉ  h  h     .. .

   h 



`id

              x = x,             

for all x

75

∈ X and for all h ∈ Gn .

(13.5)

13 Multiactions and G-spaces

(n)

The multiaction ρk

is transitive, if any two points x and y in X can be

(n)

“connected” by ρk , i.e. there exist g1 , . . . , gk

∈Gn such that

    g     1   . (n) .. x = y. ρk       g  

(13.6)

k

(n)

If g1 , . . . , gk are unique, then ρk

is sharply transitive. The subset of X, in

which any points are connected by (13.6) with fixed g1 , . . . , gk can be called the multiorbit of X. If there is only one multiorbit, then we call X the heterogenous

G-space (by analogy with the homogeneous one).

76

13 Multiactions and G-spaces

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)

     g1            ..  (n)  (n) Ψ ρk . x  = ρk             g   

k

g1 .. .

gk

     Ψ (x) ∈ Y,    

¨ which makes G-space into a category (for details, see, e.g., H USEM OLLER

(13.7)

ET AL .

[2008]). In the particular case, when X is a vector space over K, the multiaction (13.2) can be called a multi-G-module which satisfies (13.4).

77

13 Multiactions and G-spaces

The additional (linearity) conditions are

      g1          .. (n) (n) ρk . ax + by = aρk           g   k

g1 .. .

gk

          (n) x +bρk        

g1 .. .

gk

     y , (13.8)    

∈ K. Then, comparing (12.5) and (13.3) we can define a multiplace representation as a multi-G-module by the following formula       g g1  1        0 . (n,n )  .  (n) .. x . (13.9) Πk  ..  (x) = ρk          gk gk 

where a, b

In a similar way, one can generalize to polyadic systems other notions from group action theory, see e.g. K IRILLOV [1976].

78

14 Regular multiactions

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 M AL’ TCEV [1954], so called i-translations B ELOUSOV [1972]. In the binary case,

ordinary translations lead to regular representations K IRILLOV [1976], and reg(n)

therefore we call such an action a regular multiaction ρk

. In this connection,

the analog of the Cayley theorem for n-ary groups was obtained in G AL’ MAK [1986, 2001]. Now we will show in examples, how the regular multiactions can arise from i-translations.

79

14 Regular multiactions

Example 14.1. Let G3 be a ternary semigroup, k

= 2, and X =G3 , then

2-place (left) action can be defined as

  g reg(3) ρ2  h

  u def  = μ3 [g, h, u] .

(14.1)

This gives the following composition law for two regular multiactions

  g 1 reg(3) ρ2  h1

  reg(3)  g2  ρ u 2  = μ3 [g1 , h1 , μ3 [g2 , h2 , u]]  h2    μ [g , h , g ]  1 2 reg(3) 3 1 u . (14.2) = μ3 [μ3 [g1 , h1 , g2 ] , h2 , u] = ρ2   h2

Thus, using the regular 2-action ( 14.1) we have, in fact, found the associative quiver corresponding to ( 10.6).

80

14 Regular multiactions

The formula (14.1) can be simultaneously treated as a 2-translation B ELOUSOV [1972]. In this way, the following left regular multiaction

    g   1     def .. reg(n) ρk = μn [g1 , . . . , gk , h] , h .       g  

(14.3)

k

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 n0reg , which differs for even and odd values of the initial arity n.

81

14 Regular multiactions

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

n0reg = n − `id

n0reg = `μ + 1.

(14.5) (14.6)

From (14.5)–(14.6) we conclude that for any n a regular multiaction having one multiplication `μ

= 1 is binarizing and has n − 2 intact elements. For n = 3 see

(14.2). Also, it follows from (14.5) that for regular multiactions the number of intact elements gives exactly the difference between initial and final arities.

82

14 Regular multiactions

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 B OROWIEC, D UDEK ,

AND

D UPLIJ [2006], D UPLIJ [2012] to construct middle

representations of ternary groups, which did not change arity (n0

= n). Here we

give a more complicated example of a middle regular multiaction, which can contain intact elements and can therefore change arity.

83

14 Regular multiactions

Example 14.2. Let us consider 5-ary semigroup and the following middle 4-action

    g         i=3    h  ↓ reg(5)  ρ4 g, h, s , u, v  . s = μ 5  u          v  

Using ( 14.6) we observe that there are two possibilities for the number of multiplications `μ

= 2, 4. The last case `μ = 4 is similar to the vertical

84

(14.7)

14 Regular multiactions

associative quiver ( 11.4), but with a more complicated l.h.s., that is

    [g , h , g , h g ] μ   1 1 2 2, 3 5       μ [h , g , h , g , h ]   3 4 4 5 5 reg(5) 5 ρ4 s =  μ5 [u5 , v5 , u4 , v4 , u3 ]          μ [v , u , v , u , v ]   2 2 1 1 5 3       g1 g2              h  h  1 reg(5) 2 reg(5) reg(5) ρ4 ρ4 ρ4    u1 u2              v  v  1 2

85

g3 h3 u3 v3

       reg(5) ρ4      

g4 h4 u4 v4

       reg(5) ρ4       (14.8)

g5 h5 u5 v5

               s                  

14 Regular multiactions

Now we have an additional case with two intact elements `id and two multiplications `μ = 2 as reg(5) ρ4

            

i h μ5 g1 , h1 , g2 , h2, g3 h3

μ5 [h3 , v3 , u2 , v2 , u1 ] v1

       reg(5) s = ρ4      

            

g1 h1 u1 v1

reg(5) ρ4

            

g2 h2 u2 v2

reg(5) ρ4

            

g3 h3 u3 v3

= 5 to n0reg = 3. In addition to ( 14.9) we have 3 more possible regular multiactions due to the associativity of μ5 , when the with arity changing from n

multiplication brackets in the sequences of 6 elements in the first two rows and the second two ones can be shifted independently.

86

            , s               (14.9)

14 Regular multiactions

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 n0reg = 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 K IRILLOV [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 Z EKOVI C´ A RTAMONOV [1999, 2002]).

87

AND

14 Regular multiactions

This means that any element of KGn can be uniquely presented in the form

P

w = l al ∙ hl , al ∈ K, hl ∈ G. Then, using (14.3), we define the i-regular k -place representation D UPLIJ [2012]   g  1  X reg(i)  .  Πk al ∙ μk+1 [g1 . . . gi−1 hl gi+1 . . . gk ] . (14.10)  ..  (w) =   l gk 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 = k2 + 1 is called a middle regular representation. The case k = 2 was considered in B OROWIEC, D UDEK ,

AND

D UPLIJ [2006], D UPLIJ [2012] for ternary groups.

88

15 Matrix representations of ternary groups

15 Matrix representations of ternary groups Here we give several examples of matrix representations for concrete ternary groups B OROWIEC, D UDEK ,

AND

D UPLIJ [2006], D UPLIJ [2012].

= 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

Let G

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 R L L L Πreg (0, 0) = Πreg (2, 2) = Πreg (1, 1) = Πreg (0, 0)  1 0 0  R R = Πreg (2, 2) = Πreg (1, 1) =  1 0  0 0 0 1

89



  = [1] ⊕ [1] ⊕ [1], 

(15.2)

15 Matrix representations of ternary groups



 R R R L L L Πreg (2, 0) = Πreg (1, 2) = Πreg (0, 1) = Πreg (2, 1) = Πreg (1, 0) = Πreg (0, 2) =   

  = [1] ⊕  

1

− √2 3 2





3



2 1 2

  = [1] ⊕  

1

− 2 √ 3 − 2

3

2 1 − 2

0

0

0

1

1

0

0

 1  1  1 √  1 √    = [1] ⊕ − + i 3 ⊕ − − i 3 ,  2 2 2 2



 1  1  1 √  1 √    = [1] ⊕ − − i 3 ⊕ − + i 3 .  2 2 2 2

90

   

(15.3)

 R R R L L L Πreg (2, 1) = Πreg (1, 0) = Πreg (0, 2) = Πreg (2, 0) = Πreg (1, 2) = Πreg (0, 1) =   

1







0

0

0

1

1

0

0

0

1

0

(15.4)

   

15 Matrix representations of ternary groups

Consider next the middle representation construction. The middle regular representation is defined by

ΠM reg (g1 , g2 ) t =

n X

ki [g1 hi g2 ] .

i=1

For regular representations we have R R M ΠM reg (g1 , h1 ) ◦ Πreg (g2 , h2 ) = Πreg (h2 , h1 ) ◦ Πreg (g1 , g2 ) , L L M ΠM (g , h ) ◦ Π (g , h ) = Π (g , g ) ◦ Π 1 1 2 2 1 2 reg reg reg reg (h2 , h1 ) .

91

(15.5) (15.6)

15 Matrix representations of ternary groups

For the middle regular representation matrices we obtain



1

 (0, 0) = (1, 2) = (2, 1) =   0 0  0  M M M Πreg (0, 1) = Πreg (1, 0) = Πreg (2, 2) =   1 0  0  M M M Πreg (0, 2) = Πreg (2, 0) = Πreg (1, 1) =   0 1

ΠM reg

ΠM reg

ΠM reg

The above representation ΠM reg of hZ3 , [

92

0 0 1 1 0 0 0 1 0

0



 1  , 0  0  0  , 1  1  0  . 0

]i is equivalent to the orthogonal direct

15 Matrix representations of ternary groups

sum of two irreducible representations

ΠM reg

(0, 0) =

ΠM reg

(1, 2) =

ΠM reg



(2, 1) = [1] ⊕ 

−1 0

0 1



,

√  1 3 −   √ M M 2 2 ΠM (0, 1) = Π (1, 0) = Π (2, 2) = [1] ⊕ ,  reg reg reg 3 1 − − 2 2 √   1 3   √2 M M 2 (0, 2) = Π (2, 0) = Π (1, 1) = [1] ⊕ ΠM ,  reg reg reg 3 1 − 2 2 

i.e. one-dimensional trivial [1] and two-dimensional irreducible. Note, that in this

ΠM (g, g) = ΠM (g, g) 6= idV , but ΠM (g, h) ◦ ΠM (g, h) = idV , and so ΠM are of the second degree. example

93

15 Matrix representations of ternary groups

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



1

  2  [g, h, 0] =   3  0  3   0  [g, h, 2] =   1  2

2

3

0





 3 0 1    0 1 2   1 2 3  0 1 2  1 2 3    2 3 0   3 0 1

2

  3  [g, h, 1] =   0  1  0   1  [g, h, 3] =   2  3 94

3

0

1



 0 1 2    1 2 3   2 3 0  1 2 3  2 3 0    3 0 1   0 1 2

15 Matrix representations of ternary groups

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 Pn L is defined by the expansion Πreg (g1 , g2 ) t = i=1 ki [g1 g2 hi ], which means

that (see the general formula (14.10))

ΠL reg (g, h) |u >= | [ghu] > . Analogously, for right and middle representations M ΠR reg (g, h) |u >= | [ugh] >, Πreg (g, h) |u >= | [guh] > .

Therefore

| [ghu] >= | [ugh] >= | [guh] > and R M ΠL (g, h) = Π (g, h) |u >= Π reg reg reg (g, h) |u >,

so ΠL reg (g, h)

M = ΠR reg (g, h) = Πreg (g, h). Thus it is sufficient to consider

the left representation only.

95

15 Matrix representations of ternary groups

In this case the equivalence is

ΠL (a, b) = ΠL (c, d) ⇐⇒ (a + b) = (c + d)mod 4, and we obtain the

following classes



   L L L L Πreg (0, 0) = Πreg (1, 3) = Πreg (2, 2) = Πreg (3, 1) =    

   L L L L Πreg (0, 1) = Πreg (1, 0) = Πreg (2, 3) = Πreg (3, 2) =   



   L L L L Πreg (0, 2) = Πreg (1, 1) = Πreg (2, 0) = Πreg (3, 3) =    

   L L L L Πreg (0, 3) = Πreg (1, 2) = Πreg (2, 1) = Πreg (3, 0) =   

0

0

0

1

1

0

0

0

0

1

0

0

0

0

1

0

0

0

1

0

0

0

0

1

1

0

0

0

0

1

0

0

0

1

0

0

0

0

1

0

0

0

0

1

1

0

0

0

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1



    = [1] ⊕ [−1] ⊕ [−i] ⊕ [i] ,  



    = [1] ⊕ [−1] ⊕ [−1] ⊕ [−1] ,   

    = [1] ⊕ [−1] ⊕ [i] ⊕ [−i] ,  



    = [1] ⊕ [−1] ⊕ [1] ⊕ [1] .  

Due to the fact that the ternary operation (15.7) is commutative, there are only one-dimensional irreducible left representations.

96

15 Matrix representations of ternary groups

Let us “algebralize” the regular representations D UPLIJ [2012]. From (7.15) we have, for the left representation L L ΠL reg (i, j) ◦ Πreg (k, l) = Πreg (i, [jkl]) ,

where

(15.8)

L [jkl] = j − k + l, i, j, k, l ∈ Z3 . Denote γ L = Π reg (0, i), i ∈ Z3 , i

then we obtain the algebra with the relations

L L γL i γ j = γ i+j .

Conversely, any matrix representation of

(15.9)

γ i γ j = γ i+j leads to the left

ΠL (i, j) = γ j−i . In the case of the middle regular M representation we introduce γ M k+l = Πreg (k, l), k, l ∈ Z3 , then we obtain representation by

M M M γM i γ j γ k = γ [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) = γ k+l .

97

16 Polyadic rings and fields

16 Polyadic rings and fields Polyadic distributivity Let us consider a polyadic structure with two operations on the same set R: the

: Rn → R and the additional m-ary → R, that is hR | μn , ν m i. If there are no relations

“chief” (multiplication) n-ary operation μn

: Rm between μn and ν m , then nothing new, as compared with the polyadic structures having a single operation hR | μn i or hR | ν m i, can be said.

operation ν m

Informally, the “interaction” between operations can be described using the important relation of distributivity (an analog of a ∙ (b + c)

98

= a ∙ b + a ∙ c).

16 Polyadic rings and fields

Definition 16.1. The polyadic distributivity for the operations μn and ν m (no additional properties are implied for now) consists of n relations

μn [ν m [a1 , . . . am ] , b2 , b3 , . . . bn ] = ν m [μn [a1 , b2 , b3 , . . . bn ] , μn [a2 , b2 , b3 , . . . bn ] , . . . μn [am , b2 , b3 , . . . bn ]] (16.1)

μn [b1 , ν m [a1 , . . . am ] , b3 , . . . bn ] = ν m [μn [b1 , a1 , b3 , . . . bn ] , μn [b1 , a2 , b3 , . . . bn ] , . . . μn [b1 , am , b3 , . . . bn ]] (16.2) .. .

μn [b1 , b2 , . . . bn−1 , ν m [a1 , . . . am ]] = ν m [μn [b1 , b2 , . . . bn−1 , a1 ] , μn [b1 , b2 , . . . bn−1 , a2 ] , . . . μn [b1 , b2 , . . . bn−1 , am ]] , (16.3) where ai , bj

∈ R. 99

16 Polyadic rings and fields

It is seen that the operations μn and ν m enter above in a non-symmetric way, which allows us to distinguish them: one of them (μn , the n-ary multiplication) “distributes” over the other one ν m , and therefore ν m is called the addition. If only some of the relations (16.1)-(16.3) hold, then such distributivity is partial (the analog of left and right distributivity in the binary case). Remark 16.2. The operations μn and ν m need have nothing to do with ordinary multiplication (in the binary case denoted by μ2 binary case denoted by ν 2

=⇒ (∙)) and addition (in the

=⇒ (+)).

= R, n = 2, m = 3, and μ2 [b1 , b2 ] = bb12 , ν 3 [a1 , a2 , a3 ] = a1 a2 a3 (ordinary product in R). The partial distributivity now b b b b is (a1 a2 a3 ) 2 = a12 a22 a32 (only the first relation ( 16.1) holds). Example 16.3. Let A

100

16 Polyadic rings and fields

Let both operations μn and ν m be (totally) associative, which (in our definition D UPLIJ [2012]) means independence of the composition of two operations under placement of the internal operations (there are n and m such placements and therefore (n + m) corresponding relations)

μn [a, μn [b] , c] = invariant,

(16.4)

ν m [d, ν m [e] , f ] = invariant,

(16.5)

where the polyads a, b, c, d, e, f have corresponding length, and then both

hR | μn | associ and hR | ν m | associ are polyadic semigroups Sn and Sm . A commutative semigroup hA

| ν m | assoc, commi is defined by ν m [a] = ν m [σ ◦ a], for all σ ∈ Sn , where Sn is the symmetry group. If the equation ν m [a, x, b]

hR | ν m

= c is solvable for any place of x, then | assoc, solvi is a polyadic group Gm .

101

16 Polyadic rings and fields

Definition 16.4. A polyadic (m, n)-ring Rm,n is a set R with two operations

μn : Rn → R and ν m : Rm → R, such that: 1) they are distributive (16.1)-(16.3); 2) hR

| μn | associ is a n-ary semigroup;

3) hR

| ν m | assoc, comm, solvi is a commutative m-ary group.

It is obvious that a (2, 2)-ring R2,2 is an ordinary (binary) ring. Polyadic rings have much richer structure and unusual properties C ELAKOSKI ˇ UPONA [1965], L EESON [1977], C ROMBEZ [1972], C

AND

B UTSON [1980].

If the multiplicative semigroup hR

| μn | associ is commutative, μn [a] = μn [σ ◦ a], for all σ ∈ Sn , then Rm,n is called a commutative polyadic ring.

If it contains the identity, then Rm,n is a (m, n)-semiring. If the distributivity is only partial, then Rm,n is called a polyadic near-ring. 102

16 Polyadic rings and fields

A polyadic ring is derived, if ν m and μn are equivalent to a repetition of the binary addition and multiplication, while hR (binary) group and semigroup.

| +i and hR | ∙i are commutative

An n-admissible “length of word (x)” should be congruent to 1 mod (n − 1), containing `μ (n − 1) + 1 elements (`μ is a “number of multiplications”) (` )

μn μ [x] (x ∈ R`μ (n−1)+1 ), or polyads. An m-admissible “quantity of words (y)” in a polyadic “sum” has to be congruent to 1 mod (m − 1), i.e. consisting (`ν ) of `ν (m − 1) + 1 summands (`ν is a “number of additions”) ν m [y] (y ∈ R`ν (m−1)+1 ). “Polyadization” of a binary expression (m

= n = 2): the multipliers `μ + 1 → `μ (n − 1) + 1 and summands `ν + 1 → `ν (m − 1) + 1.

Example 16.5. “Trivial polyadization”: the simplest (m, n)-ring derived from the

ring of integers Z as the set of `ν

(m − 1) + 1 “sums” of n-admissible (`μ (n − 1) + 1)-ads (x), x ∈ Z`μ (n−1)+1 L EESON AND B UTSON [1980].

103

16 Polyadic rings and fields

The additive m-ary polyadic power and multiplicative n-ary polyadic power are k

z }| { (inside polyadic products we denote repeated entries by x, . . . , x as xk ) h i h i h`μ i×n h`ν i+m (`ν ) `ν (m−1)+1 (`μ ) `μ (n−1)+1 , x x , x ∈ R, x = νm x = μn (16.6)

Polyadic and ordinary powers differ by 1:

xh`ν i+2 = x`ν +1 , xh`μ i×2 = x`μ +1 .

The polyadic idempotents in Rm,n satisfy

xh`ν i+m = x,

xh`μ i×n = x,

(16.7)

and are called the additive `ν -idempotent and the multiplicative `μ -idempotent. The idempotent zero z

∈ R, is (if it exists) defined by ν m [x, z] = z, ∀x ∈ Rm−1 .

(16.8)

If a zero z exists, it is unique. An element x is nilpotent, if

xh1i+m = z. 104

(16.9)

16 Polyadic rings and fields

The unit e of Rm,n is multiplicative 1-idempotent



μn e

n−1



∀x ∈ R.

, x = x,

(16.10)

In case of a noncommutative polyadic ring x can be on any place. In distinction with the binary case there are unusual polyadic rings : 1) with no unit and no zero (zeroless, nonunital); 2) with several units and no zero; 3) with all elements are units. In polyadic rings invertibility is not connected with unit and zero elements. For a fixed element x

∈ R its additive querelement x ˜ and multiplicative

ˉ are defined by querelement x 

νm x Because hR

m−1





,x ˜ = x,

μn x

n−1



,x ˉ = x,

| ν m i is a commutative group, each x ∈ R has its additive querelement x ˜ (and is querable or “polyadically invertible”). 105

(16.11)

16 Polyadic rings and fields

The n-ary semigroup hR If each x

| μn i can have no multiplicatively querable elements.

∈ R has its unique querelement, then hR | μn i is an n-ary group.

Denote R∗ If hR∗

= R \ {z}, if the zero z exists.

| μn i is the n-ary group, then Rm,n is a (m, n)-division ring. Definition 16.6. A commutative (m, n)-division ring is a (m, n)-field Fm,n . Example 16.7. a) The set iR with i2 = −1 is a (2, 3)-field with no unit and a zero 0 (operations in C), the multiplicative querelement of ix is −ix (x 6= 0).  odd 2 b) The set of fractions ix/y | x, y ∈ Z , i = −1 is a (3, 3)-field with no zero and no unit (operations are in C), while the additive and multiplicative querelements of ix/y are −ix/y and −iy/x, respectively. c) The set ofantidiagonal 2 ×2 matrices over R is a (2, 3)-field with zero  

z = 

 

0

0

0

0

0

x

y

0





, two units e = ± 



 is 

0

1/y

1/x

0

0

1

1

0





, but the multiplicative querelement of

. 106

17 Polyadic analogs of integer number ring Z and field Z/pZ

17 Polyadic analogs of integer number ring Z and field Z/pZ The theory of finite fields L IDL

AND

N IEDERREITER [1997] plays a very important

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

107

17 Polyadic analogs of integer number ring Z and field Z/pZ

We define new secondary congruence classes and the corresponding finite

(m, n)-rings Z(m,n) (q) of polyadic integer numbers, which give ZqZ in the “binary limit”. We construct the prime polyadic fields F(m,n) (q), which can be treated as polyadic analog of the Galois field GF (p). Ring of polyadic integer numbers Z(m,n) The ring of polyadic integer numbers Z(m,n) was introduced in D UPLIJ [2017a]. Consider a congruence class (residue class) of an integer a modulo b

[[a]]b = {{a + bk} | k ∈ Z, a ∈ Z+ , b ∈ N, 0 ≤ a ≤ b − 1} . Denote a representative by xk

[a,b]

= xk

(17.1)

= a + bk , where {xk } is an infinite set.

108

17 Polyadic analogs of integer number ring Z and field Z/pZ

Informally, there are two ways to equip (17.1) with operations: 1. The “External” way: to define operations between the classes [[a]]b . Denote the class representative by [[a]]b

+0 , ∙0 as

≡ a0 , and introduce the binary operations 0

a01 +0 a02 = (a1 + a2 ) , 0

a01 ∙0 a02 = (a1 a2 ) .

(17.2) (17.3)

The binary residue class ring is defined by

ZbZ = {{a0 } | +0 , ∙0 , 00 , 10 } . With prime b

(17.4)

= p, the ring ZpZ is a binary finite field having p elements.

This is the standard finite field theory L IDL

109

AND

N IEDERREITER [1997].

17 Polyadic analogs of integer number ring Z and field Z/pZ

2. The “Internal” way is to introduce (polyadic) operations inside a class [[a]]b (with both a and b fixed). Define the commutative m-ary addition and commutative n-ary multiplication of representatives xki in [[a]]b by

ν m [xk1 , xk2 , . . . , xkm ] = xk1 + xk2 + . . . + xkm ,

(17.5)

μn [xk1 , xk2 , . . . , xkn ] = xk1 xk2 . . . xkn , xki ∈ [[a]]b , ki ∈ Z.

(17.6)

Remark 17.1. Binary sums xk1

+ xk2 and products xk1 xk2 are not in [[a]]b for arbitrary a ∈ Z+ , b ∈ N, 0 ≤ a ≤ b − 1.

110

17 Polyadic analogs of integer number ring Z and field Z/pZ

Proposition 17.2 ( D UPLIJ [2017b]). The polyadic operations ν m and μn closed in [[a]]b , if the arities (m, n) have the minimal values satisfying

ma ≡ a (mod b) ,

(17.7)

an ≡ a (mod b) .

Remark 17.3. If n

(17.8)

= b = p is prime, then (17.8) is valid for any a ∈ N,

which is another formulation of Fermat’s little theorem.

Definition 17.4 ( D UPLIJ [2017a]). The congruence class [[a]]b equipped with a structure of nonderived infinite commutative polyadic ring is called a

(m, n)-ring of polyadic integer numbers [a,b]

Z(m,n) ≡ Z(m,n) = {[[a]]b | ν m , μn } .

(17.9)

Definition 17.5. A polyadic prime number is such that obeys a unique expansion

h

i

xkp = μn(`) xkp , e`(n−1) , where e a polyadic unit of Z(m,n) (if exists).

111

(17.10)

17 Polyadic analogs of integer number ring Z and field Z/pZ

Example 17.6. In the residue class

[[3]]4 = {. . . − 25, −21, −17, −13, −9, −5, −1, 3, 7, 11, 15, 19, 23, 27, 31 . . .} (17.11)

To retain the same class [[3]]4 , we can add 4`ν

+ 1 = 5, 9, 13, 17, . . . representatives and multiply 2`μ + 1 = 3, 5, 7, 9, . . . representatives only. E.g., take “number” of additions `ν

= 2, and multiplications `μ = 3, to get (7 + 11 + 15 + 19 + 23) − 5 − 9 − 13 − 17 = 31 = 3 + 4 ∙ 7 ∈ [[3]]4 , ((7 ∙ 3 ∙ 11) ∙ 19 ∙ 15) ∙ 31 ∙ 27 = 55 103 895 = 3 + 4 ∙ 13775973 ∈ [[3]]4 . This means that [[3]]4 is the polyadic (5, 3)-ring Z(5,3)

[3,4]

= Z(5,3) .

Remark 17.7. Elements of the (m, n)-ring Z(m,n) (polyadic integer numbers) [a,b]

are not ordinary integers (forming a (2, 2)-ring). A representative xk [3,4]

, e.g.

[1,2]

3 = 3(5,3) ∈ Z(5,3) is different from 3 = 3(3,2) ∈ Z(3,2) , and different from the binary 3

[0,1]

∈ Z ≡ Z(2,2) , i.e. properly speaking 3(5,3) 6= 3(3,2) 6= 3, since their

operations (multiplication and addition) are different.

112

17 Polyadic analogs of integer number ring Z and field Z/pZ

The parameters-to-arity mapping Remark 17.8. a) Solutions to (17.7) and (17.8) do not exist for all a and b; b) The pair a, b determines m, n uniquely; c) For several different pairs a, b there can be the same arities m, n. Assertion 17.9. The parameters-to-arity mapping ψ partial surjection.

: (a, b) −→ (m, n) is a

The characterization of the fixed congruence class [[a]]b and the corresponding [a,b]

(m, n)-ring of polyadic integer numbers Z(m,n) can be done in terms of the shape invariants I, J ∈ Z+ defined uniquely by (TABLE 3 in D UPLIJ [2017a]) I=

[a,b] Im

a = (m − 1) , b

J=

In the binary case, when m

Jn[a,b]

an − a = . b

(17.12)

= n = 2 (a = 0, b = 1), both shape invariants vanish, I = J = 0. There exist “partially” binary cases, when only n = 2 and [6,10] m 6= 2, while J is nonzero, for instance in Z(6,2) we have I = J = 3. 113

17 Polyadic analogs of integer number ring Z and field Z/pZ

Polyadic rings of secondary classes A special method of constructing a finite nonderived polyadic ring by combining the “External” and “Internal” methods was given in D UPLIJ [2017b]. Introduce the finite polyadic ring Z(m,n) cZ, where Z(m,n) is a polyadic ring. If we directly consider the “double” class {a + bk

+ cl} and fix a and b, then the factorization by cZ will not give a closed operations for arbitrary c. [a,b] Assertion 17.10. If the finite polyadic ring Z(m,n) cZ has q elements, then c = bq.

(17.13)

Definition 17.11. A secondary (equivalence) class of a polyadic integer [a,b]

xk

[a,b]

= a + bk ∈ Z(m,n) “modulo” bq (with q being the number of [a,b]

representatives xk

hh

[a,b] xk

ii

bq

, for fixed b

∈ N and 0 ≤ a ≤ b − 1) is

= {{(a + bk) + bql} | l ∈ Z, q ∈ N, 0 ≤ k ≤ q − 1} .

(17.14)

114

17 Polyadic analogs of integer number ring Z and field Z/pZ

Remark 17.12. In the binary limit a

[0,1]

= 0, b = 1 and Z(2,2) = Z, the secondary

class becomes the ordinary class (17.1). If the values of a, b, q are clear from the context, hh we denote ii the secondary class representatives by an integer with two primes

Example 17.13. a) For a

hh

[3,6] xk

b) If a

ii

24

[a,b]

xk

bq

≡ x00k ≡ x00 .

= 3, b = 6 and for 4 elements and k = 0, 1, 2, 3

= 300 , 900 , 1500 , 2100 ,

([[k]]4 = 00 , 10 , 20 , 30 ) . (17.15)

= 4, b = 5, for 3 elements and k = 0, 1, 2 we get hh ii [4,5] xk = 400 , 900 , 1400 , ([[k]]3 = 00 , 10 , 20 ) . 15

115

(17.16)

17 Polyadic analogs of integer number ring Z and field Z/pZ

c) Let a

= 3, b = 5, then for q = 4 elements we have the secondary classes with k = 0, 1, 2, 3 (the binary limits are in brackets)  00  = {. . . − 17, 3, 23, 43, 63, . . .} , 3      800 = {. . . − 12, 8, 28, 48, 68, . . .} , hh ii [3,5] 00 00 00 00 xk = 3 , 8 , 13 , 18 = 00  20 = {. . . − 7, 13, 33, 53, 73, . . .} , 13      1800 = {. . . − 2, 18, 38, 58, 78, . . .} , (17.17)

 0  = {. . . − 4, 0, 4, 8, 12, . . .} , 0        10 = {. . . − 3, 1, 5, 9, 13, . . .} ,  [[k]]4 = 00 , 10 , 20 , 30 = 0   = {. . . − 2, 2, 6, 10, 14, . . .} , 2       30 = {. . . − 1, 3, 7, 11, 15, . . .} . 

(17.18)

Difference between classes: 1) they are described by rings of different arities; 2) some of them are fields. 116

      

17 Polyadic analogs of integer number ring Z and field Z/pZ

Finite polyadic rings Now we determine the nonderived polyadic operations between secondary classes which lead to finite polyadic rings. Proposition 17.14. The set {x00 k } of q secondary classes k

= 0, . . . , q − 1

(with the fixed a, b) can be endowed with the commutative m-ary addition

x00kadd

=

ν 00m 



x00k1 , x00k1 , . . . , x00km



,

(17.19)



[a,b] kadd ≡ (k1 + k2 + . . . + km ) + Im (mod q)

and commutative n-ary multiplication

x00kmult

=

μ00n



x00k1 , x00k1 , . . . , x00kn



,

(17.20)

(17.21)

kmult ≡ an−1 (k1 + k2 + . . . + kn ) + an−2 b (k1 k2 + k2 k3 + . . . + kn−1 kn ) + . . .  (17.22) +bn−1 k1 . . . kn + Jn[a,b] (mod q) , [a,b]

satisfying the polyadic distributivity, shape invariants Im

117

[a,b]

, Jn

are in ( 17.12).

17 Polyadic analogs of integer number ring Z and field Z/pZ

Definition 17.15. The set of secondary classes (17.14) equipped with operations (17.19), (17.21) is denoted by [a,b]

[a,b]

Z(m,n) (q) ≡ Z(m,n) (q) = Z(m,n)  (bq) Z = {{x00k } | ν 00m , μ00n } , (17.23) and is a finite secondary class (m, n)-ring of polyadic integer numbers [a,b]

Z(m,n) ≡ Z(m,n) . The value q (the number of elements) is called its order. [3,4]

Example 17.16. a) In (5, 3)-ring Z(4,3) (2) with 2 secondary classes all

elements are units (marked by subscript e) e1

= 300e = 300 , e2 = 700e = 700 ,

because

μ3 [300 , 300 , 300 ] = 300 , μ3 [300 , 300 , 700 ] = 700 , μ3 [300 , 700 , 700 ] = 300 , μ3 [700 , 700 , 700 ] = 700 . (17.24) [5,6]

b) The ring Z(7,3) (4) consists of 4 units

e1 = 500e , e2 = 1100e , e3 = 1700e , e4 = 2300e , and no zero. [a1 ,b1 ]

Remark 17.17. Equal arity finite polyadic rings of the same order Z(m,n) [a2 ,b2 ]

and Z(m,n)

(q) may be not isomorphic.

118

(q)

17 Polyadic analogs of integer number ring Z and field Z/pZ

[1,3]

Example 17.18. The finite polyadic ring Z(4,2) (2) of order 2 consists of unit

e = 100e = 100 and zero z = 400z = 400 only,

μ2 [100 , 100 ] = 100 , μ2 [100 , 400 ] = 400 , μ2 [400 , 400 ] = 400 ,

(17.25)

[1,3]

00 and therefore Z(4,2) (2) is a field, because {100 , 400 z } \ 4z is a (trivial) binary

group, consisting of one element 100 e. [4,6]

However, Z(4,2) (2) has the zero z

= 400z = 400 , 1000 and has no unit, because

μ2 [400 , 400 ] = 400 , μ2 [400 , 1000 ] = 400 , μ2 [1000 , 1000 ] = 400 ,

(17.26)

[4,6]

so that Z(4,2) (2) is not a field, because the nonzero element 1000 is nilpotent. [1,3]

Their additive 4-ary groups are also not isomorphic, while Z(4,2) (2) and [4,6]

Z(4,2) (2) have the same arity (m, n) = (4, 2) and order 2. Assertion 17.19. For a fixed arity shape (m, n), there can be non-isomorphic secondary class polyadic rings Z(m,n) (q) of the same order q , which describe different binary residue classes [[a]]b . 119

17 Polyadic analogs of integer number ring Z and field Z/pZ

Finite polyadic fields [a,b]

Proposition 17.20. A finite polyadic ring Z(m,n) (q) is a secondary class finite 00[a,b]

(m, n)-field F(m,n) (q) if all its elements except z (if it exists) are polyadically multiplicative invertible having a unique querelement. In the binary case L IDL

AND

N IEDERREITER [1997] the residue (congruence)

class ring (17.4) with q elements ZqZ is a congruence class (non-extended) field, if its order nn q

F0 (p) =

such that =op is a prime number, o [[a]]p | +0 , ∙0 , 00 , 10 , a = 0, 1, . . . , p − 1.

All non-extended binary fields of a fixed prime order p are isomorphic, and   so it is

natural to study them in a more “abstract” way. The mapping Φp an isomorphism of binary fields Φp

: F0 (p) → F (p), where

[[a]]p = a is

F (p) = {{a} | +, ∙, 0, 1}mod p is an “abstract” non-extended (prime) finite field of order p (or Galois field GF (p)).

120

17 Polyadic analogs of integer number ring Z and field Z/pZ

Consider n the set o of polyadic integer numbers [a,b]

[a,b]

{xk } ≡ xk = {a + bk} ∈ Z(m,n) , b ∈ N and 0 ≤ a ≤ b − 1, 0 ≤ k ≤ q − 1, q ∈ N, which obey the operations

(17.5)–(17.6).

Definition 17.21. The “abstract” non-extended finite (m, n)-field of order q is [a,b]

F(m,n) (q) ≡ F(m,n) (q) = {{a + bk} | ν m , μn }mod bq ,

(17.27)

if {{xk }

| ν m }mod bq is an additive m-ary group, and {{xk } | μn }mod bq (or, when zero z exists, {{xk \ z} | μn }mod bq ) is a multiplicative n-ary group.

Define a one-to-one ontomapping from  the secondary congruence class to its [a,b] representative by Φq

hh

[a,b] xk

ii

[a,b]

bq [a,b] Proposition 17.22. The mapping Φq

= xk

and arrive

00[a,b]

[a,b]

: F(m,n) (q) → F(m,n) (q) is a

polyadic ring homomorphism (being, in fact, an isomorphism).

In TABLE 2 we present the “abstract” non-extended polyadic finite fields [a,b]

F(m,n) (q) of lowest arity shape (m, n) and orders q .

121

17 Polyadic analogs of integer number ring Z and field Z/pZ

[a,b]

[a,b]

Table 2: The finite polyadic rings Z(m,n) (q) and (m, n)-fields Fm,n a\b

2

3

4

m = 3

m = 4

m = 5

m = 6

n = 2

n = 2

n = 2

n = 2

1e ,3

1e ,4z

1e ,5

5

(q). 6

m = 7 n = 2

1e ,6z

1e ,7

1 1e ,3z ,5

1e ,4,7

1e ,5,9z

1e ,3,5,7

1e ,4z ,7,10

1e ,5,9,13

q=5,7,8

q=5,7,9

q=5,7,8

1e ,6z ,11

1e ,7,13 1e ,7,13,19

1e ,6,11,16z

q=5,6,7,8,9

q=5,7

m = 6

m = 4

n = 5

n = 3

m = 4 n = 3

2z ,7e

2z ,5e 2 2,5,8e

2e ,7,12z

2,8z 2,8e ,14 2,8z ,14,20

2,5e ,8z ,11e

2,7e ,12z ,17e

q=5,7,9

q=5,7,9

q=5,7

m = 5

m = 6

n = 3

n = 5

3e ,7e

m = 3 n = 2

3e ,8z

3,9e

3 3z ,7e ,11e 3,7e ,11,15e

122q=5,6,7,8

3z ,8e ,13e

3,9z ,15 3,9e ,15,21

3e ,8z ,13e ,18 q=5,7

q=5,7,8

17 Polyadic analogs of integer number ring Z and field Z/pZ

In the multiplicative structure the following crucial differences between the binary finite fields F (q) and polyadic fields F(m,n) (q) can be outlined. 1. The order of a non-extended finite polyadic field may not be prime (e.g., [1,2]

[3,4]

[2,6]

F(3,2) (4), F(5,3) (8), F(4,3) (9)), and may not even be a power of a prime binary number (e.g.

[5,6]

[3,10]

F(7,3) (6), F(11,5) (10)), and see TABLE 3.

2. The polyadic characteristic χp of a non-extended finite polyadic field can have values such that χp

+ 1 (corresponding in the binary case to the

ordinary characteristic χ) can be nonprime. 3. There exist finite polyadic fields with more than one unit, and also all elements can be units. Such cases are marked in TABLE 3 by subscripts which indicate the number of units. 4. The(m, n)-fields can be zeroless-nonunital, but have unique additive and multiplicative querelements:

h[[a]]b | ν m i, h[[a]]b | μn i are polyadic groups.

5. The zeroless-nonunital polyadic fields are totally (additively and multiplicatively) nonderived. 123

17 Polyadic analogs of integer number ring Z and field Z/pZ

Example 17.23. 1) The zeroless-nonunital polyadic finite fields having lowest [3,8]

[3,8]

[5,8]

[5,8]

[4,9]

|a + b| are, e.g., F(9,3) (2), F(9,3) (4), F(9,3) (4), F(9,3) (8), also F(10,4) (3), [4,9]

[7,9]

[7,9]

F(10,4) (9), and F(10,4) (3), F(10,4) (9).

[5,8]

2) The multiplication of the zeroless-nonunital (9, 3)-field F(9,3) (2) is

μ3 [5, 5, 5] = 13, μ3 [5, 5, 13] = 5, μ3 [5, 13, 13] = 13, μ3 [13, 13, 13] = 5. The (unique) multiplicative querelements ˉ 5

= 13, 13 = 5. The addition is

i i h i h h i h h i 5 4 6 3 8 7 2 9 = 13, = 5, ν 9 5 , 13 = 13, ν 9 5 , 13 = 13, ν 9 5 , 13 = 5, ν 9 5 , 13 ν9 5 i i h i h i h i h h 9 8 2 7 3 6 4 5 = 5. = 13, ν 9 13 = 5, ν 9 5, 13 = 13, ν 9 5 , 13 = 5, ν 9 5 , 13 ν 9 5 , 13

= 13, f 13 = 5. So all elements are additively and multiplicatively querable (polyadically invertible), and therefore ν 9 is 9-ary additive group operation and μ3 is 3-ary multiplicative group operation, [5,8] as it should be for a field. Because it contains no unit and no zero, F(9,3) (2) is actually a zeroless-nonunital finite (9, 3)-field of order 2. The additive (unique) querelements are ˜ 5

124

17 Polyadic analogs of integer number ring Z and field Z/pZ

[2,3]

Example 17.24. The (4, 3)-ring Z(4,3) (6) is zeroless, and h[[3]]4

| ν 4 i is its

4-ary additive group (each element has a unique additive querelement). Despite each element of h[[2]]3 | μ3 i having a querelement, it is not a multiplicative 3-ary group, because for the two elements 2 and 14 we have nonunique querelements μ3 [2, 2, 5] = 2, μ3 [2, 2, 14] = 2, μ3 [14, 14, 2] = 14, μ3 [14, 14, 11] = 14. (17.28) Example 17.25. The polyadic (9, 3)-fields corresponding to the congruence classes [[5]]8 and [[7]]8 are not isomorphic for orders q

= 2, 4, 8 (see TABLE 3).

[5,8]

Despite both being zeroless, the first F(9,3) (q) are nonunital, while the second [7,8]

F(9,3) (q) has two units, which makes an isomorphism impossible.

125

17 Polyadic analogs of integer number ring Z and field Z/pZ

Polyadic field order In binary case the order of an element x

∈ F (p) is defined as a smallest integer

λ such that xλ = 1. Obviously, the set of fixed order elements forms a cyclic subgroup of the multiplicative binary group of F (p), and λ | (p − 1). If λ

= p − 1, such an element is called a primitive (root), it generates all

elements, and these exist in any finite binary field. Any element of F (p) is idempotent xp

= x, while all its nonzero elements satisfy

xp−1 = 1 (Fermat’s little theorem). A non-extended (prime) finite field is fully determined by its order p (up to isomorphism), and, moreover, any F (p) is isomorphic to ZpZ. In the polyadic case, the situation is more complicated. Because the related secondary class structure (17.27) contains parameters in addition to the number of elements q , the order of (non-extended) polyadic fields may not be prime, or nor even a power of a prime integer (e.g.

[5,6]

[3,10]

F(7,3) (6) or F(11,5) (10)).

126

17 Polyadic analogs of integer number ring Z and field Z/pZ

Because finite polyadic fields can be zeroless, nonunital and have many (or even all) units (see TABLE 3), we cannot use units in the definition of the element order. Definition 17.26. If x

∈ F(m,n) (q) satisfies xhλp i×n = x,

(17.29)

then the smallest such λp is called the idempotence polyadic order ord x [a,b]

Definition 17.27. The idempotence polyadic order λp

= λp .

of a finite polyadic field

[a,b]

F(m,n) (q) is the maximum λp of all its elements, we call such field [a,b]

[a,b] -idempotent and denote ord F(m,n) (q) = λp . λ[a,b] p [a,b]

In TABLE 3 we present the idempotence polyadic order λp

for small a, b.

Definition 17.28. Denote by q∗ the number of nonzero distinct elements in

F(m,n) (q)

  q − 1, if ∃z ∈ F (m,n) (q) q∗ =  q, if @z ∈ F(m,n) (q) ,

(17.30)

which is called a reduced (field) order (in binary case we have the first line only).

127

17 Polyadic analogs of integer number ring Z and field Z/pZ

[a,b]

Table 3: Idempotence polyadic orders λp [[a]]b

[a,b]

for finite polyadic fields F(m,n) (q).

Finite polyadic field order q

Arities

b

a

(m, n)

2

3

4

5

6

7

8

9

10

2

1

(3, 2)

2

2

2

4



6

4





3

1

(4, 2)

1

3



4



6



9



2

(4, 3)

1

3



22e



32e



9



1

(5, 2)

2

2

4

4



6

8





3

(5, 3)

12e

12e

22e

22e



32e

42e





1

(6, 2)

1

2



5



6







2

(6, 5)

1

12e



5



32e







3

(6, 5)

1

12e



5



32e







4

(6, 3)

1

12e



5



32e







1

(7, 2)

2

3

2

4

6

6

4

9



2

(4, 3)



3



22e



32e



9



3

(3, 2)

2



2

4



6

4





4

(4, 2)



3



4



6



9



5

(7, 3)

12e

3

14e

22e

32e

32e

2

9



1

(8, 2)

1

2



4



7







2

(8, 4)

1

2



4



7







3

(8, 7)

1

12e



22e



7







4

(8, 4)

1

2



4



7







5

(8, 7)

1

12e



22e



7







6

(8, 3)

1

12e



22e



7







1

(9, 2)

2

2

4

4



6

8





3

(9, 3)

2

12e



32e

8





4

5

6

7

8

4 128 22e

17 Polyadic analogs of integer number ring Z and field Z/pZ

Theorem 17.29. If a finite polyadic field F(m,n) (q) has an order q , such that

q∗ = q∗adm = ` (n − 1) + 1 is n-admissible, then (for n ≥ 3 and one unit): 1. A sequence of the length q∗ (n − 1) built from any fixed element

y ∈ F(m,n) (q) is neutral h i μn(q∗ ) x, y q∗ (n−1) = x, ∀x ∈ F(m,n) (q) .

(17.31)

2. Any element y satisfies the polyadic idempotency condition

y hq∗ i×n = y, ∀y ∈ F(m,n) (q) .

(17.32)

[a,b]

Finite polyadic fields F(m,n) (q) having n-admissible reduced order

q∗ = q∗adm = ` (n − 1) + 1 (` ∈ N) (underlined in TABLE 3) are closest to the binary finite fields F (p) in their general properties: they are always half-derived, while if they additionally contain a zero, they are fully derived. If q∗

[a,b]

6= q∗adm , then F(m,n) (q) can be nonunital or contain more than one unit

(subscripts in TABLE 3).

129

17 Polyadic analogs of integer number ring Z and field Z/pZ

[a,b]

Assertion 17.30. The finite fields F(m,n) (q) of n-admissible reduced order

q∗ = q∗adm cannot have more than one unit and cannot be zeroless-nonunital. Assertion 17.31. If q∗

[a,b]

6= q∗adm , and F(m,n) (q) is unital zeroless, then the

reduced order q∗ is the product of the idempotence polyadic field order [a,b]

λ[a,b] = ord F(m,n) (q) and the number of units κe (if a - b and n ≥ 3) p q∗ = λ[a,b] κe . p [a,b]

Structure of the multiplicative group Gn

(17.33)

[a,b]

(q∗ ) of F(m,n) (q)

Some properties of commutative cyclic n-ary groups were considered for particular relations between orders and arity. Here we have: 1) more parameters and different relations between these, the arity m, n and order q ; 2) the

(m, n)-field under consideration, which leads to additional restrictions. In such a way exotic polyadic groups and fields arise which have unusual properties that have not been studied before.

130

17 Polyadic analogs of integer number ring Z and field Z/pZ

Definition 17.32. An element xprim idempotence order is

[a,b]

∈ Gn

(q∗ ) is called n-ary primitive, if its

λp = ord xprim = q∗ .

(17.34)

h1i×n h2i×n hq∗ i×n All λp polyadic powers xprim , xprim , . . . , xprim ≡ xprim generate other [a,b] elements, and so Gn is a finite cyclic n-ary group generated by xprim , (q∗ ) o Dn hii E [a,b] ×n i.e. Gn (q∗ ) = xprim | μn . Number primitive elements in κprim . [a,b]

Assertion 17.33. For zeroless F(m,n) (q) and prime order q [a,b]

λ[a,b] = q , and Gn p

= p, we have

(q) is indecomposable (n ≥ 3).

[2,3]

Example 17.34. The smallest 3-admissible zeroless polyadic field is F(4,3) (3)

with the unit e

= 8e and two 3-ary primitive elements 2, 5 having 3-idempotence order ord 2 = ord 5 = 3, so κprim = 2 , because

2h1i×3 = 8e , 2h2i×3 = 5, 2h3i×3 = 2,

5h1i×3 = 8e , 5h2i×3 = 2, 5h3i×3 = 5, (17.35)

[2,3]

and therefore G3

(3) is a cyclic indecomposable 3-ary group. 131

17 Polyadic analogs of integer number ring Z and field Z/pZ

[a,b]

Assertion 17.35. If F(m,n) (q) is zeroless-nonunital, then every element is

n-ary primitive, κprim = q , also λ[a,b] = q (the order q can be not prime), and p [a,b]

Gn (q) is a indecomposable commutative cyclic n-ary group without identity (n ≥ 3). [5,9] Example 17.36. The (10, 7)-field F(10,7) (9) is zeroless-nonunital, each element (has λp = 9) is primitive and generates the whole field, and therefore [5,9] κprim = 9, thus the 7-ary multiplicative group G7 (9) is indecomposable and without identity.

[a,b]

The structure of Gn

in the binary case.

(q∗ ) can be extremely nontrivial and may have no analogs

Assertion 17.37. If there exists more than one unit, then: [a,b]

1. If Gn

(q∗ ) can be decomposed on its n-ary subroups, the number of units κe coincides with the number of its cyclic n-ary subgroups [a,b] Gn (q∗ ) = G1 ∪ G2 . . . ∪ Gke which do not intersect Gi ∩ Gj = ∅, i, j = i = 1, . . . , κe , i 6= j . 132

17 Polyadic analogs of integer number ring Z and field Z/pZ

2. If a zero exists, then each Gi has its own unit ei , i [a,b]

3. In the zeroless case Gn

= 1, . . . , κe .

(q) = G1 ∪ G2 . . . ∪ Gke ∪ E (G), where

E (G) = {ei } is the split-off subgroup of units. [5,8]

Example 17.38. 1) In the (9, 3)-field F(9,3) (7) there is a single zero z

= 21z

and two units e1

= 13e , e2 = 29e , and so its multiplicative 3-ary group [5,8] G3 (6) = {5, 13e , 29e , 37, 45, 53} consists of two nonintersecting (which is not possible in the binary case) 3-ary cyclic subgroups G1 = {5, 13e , 45} and G2 = {29e , 37, 53} (for both λp = 3) n o h1i×3 h2i×3 h3i×3 G1 = 5 = 13e , 5 = 45, 5 = 5 , ˉ5 = 45, 45 = 5, n o h1i×3 h2i×3 h3i×3 G2 = 37 = 29e , 37 = 53, 37 = 37 , 37 = 53, 53 = 37. [5,8]

All nonunital elements in G3

(6) are (polyadic) 1-reflections, because 5h1i×3 = 45h1i×3 = 13e and 37h1i×3 = 53h1i×3 = 29e , and so the subgroup of units E (G) = {13e , 29e } is unsplit E (G) ∩ G1,2 6= ∅. 133

17 Polyadic analogs of integer number ring Z and field Z/pZ

[7,8]

2) For the zeroless F(9,3) (8), its multiplicative 3-group [5,8]

G3 (6) = {7, 15, 23, 31e , 39, 47, 55, 63e } has two units e1 = 31e , e2 = 63e , and it splits into two nonintersecting nonunital cyclic 3-subgroups (λp = 4 and λp = 2) and the subgroup of units n o G1 = 7h1i×3 = 23, 7h2i×3 = 39, 7h3i×3 = 55, 7h4i×3 = 4 , ˉ 7 = 55, 55 = 7, 23 = 39, 39 = 23, n o G2 = 15h1i×3 = 47, 15h2i×3 = 15 , 15 = 47, 47 = 15, E (G) = {31e , 63e } .

There are no `μ -reflections, and so E (G) splits out E (G) ∩ G1,2 If all elements are units E (G) [a,b]

[a,b]

= Gn

= ∅.

(q), the group is 1-idempotent λp = 1.

Assertion 17.39. If F(m,n) (q) is zeroless-nonunital, then there no n-ary cyclic [a,b]

subgroups in Gn

(q).

134

17 Polyadic analogs of integer number ring Z and field Z/pZ

[a,b]

The subfield structure of F(m,n) (q) can coincide with the corresponding [a,b]

subgroup structure of the multiplicative n-ary group Gn

(q∗ ), only if its additive

m-ary group has the same subgroup structure. However, we have

[a,b]

Assertion 17.40. Additive m-ary groups of all polyadic fields F(m,n) (q) have the same structure: they are polyadically cyclic and have no proper m-ary subgroups. Therefore, in additive m-ary groups each element generates all other elements, i.e. it is a primitive root. [a,b]

Theorem 17.41. The polyadic field F(m,n) (q), being isomorphic to the [a,b]

(m, n)-field of polyadic integer numbers Z(m,n) (q), has no any proper subfield. [a,b]

In this sense, F(m,n) (q) can be named a prime polyadic field.

135

18 Multiplicative properties of exotic finite polyadic fields

18 Multiplicative properties of exotic finite polyadic fields We list examples of finite polyadic fields are not possible in the binary case. Only the multiplication of fields will be shown, because their additive part is huge (many pages) for higher arities, and does not carry so much distinctive information. 1) The first exotic finite polyadic field which has a number of elements which is not a prime number, or prime power (as it should be for a finite binary field) is [5,6]

F(7,3) (6), which consists of 6 elements {5, 11, 17, 23, 29, 35}, q = 6, It is zeroless and contains two units {17, 35} ≡ {17e , 35e }, κe = 2, and each element has the idempotence polyadic order λp = 3, i.e.  7 [5,6] h3i x = μ3 x = x, ∀x ∈ F(7,3) (6). 136

18 Multiplicative properties of exotic finite polyadic fields

The multiplication is μ3 [5, 5, 5] = 17, μ3 [5, 5, 11] = 23, μ3 [5, 5, 17] = 29, μ3 [5, 5, 23] = 35, μ3 [5, 5, 29] = 5, μ3 [5, 5, 35] = 11, μ3 [5, 11, 11] = 29, μ3 [5, 11, 17] = 35, μ3 [5, 11, 23] = 5, μ3 [5, 11, 29] = 11, μ3 [5, 11, 35] = 17, μ3 [5, 17, 17] = 5, μ3 [5, 17, 23] = 11, μ3 [5, 17, 29] = 17, μ3 [5, 17, 35] = 23, μ3 [5, 23, 23] = 17, μ3 [5, 23, 29] = 23, μ3 [5, 23, 35] = 29, μ3 [5, 29, 29] = 29, μ3 [5, 29, 35] = 35, μ3 [5, 35, 35] = 5, μ3 [11, 11, 11] = 35, μ3 [11, 11, 17] = 5, μ3 [11, 11, 23] = 11, μ3 [11, 11, 29] = 17, μ3 [11, 11, 35] = 23, μ3 [11, 11, 17] = 5, μ3 [11, 17, 17] = 11, μ3 [11, 17, 23] = 17, μ3 [11, 17, 29] = 23, μ3 [11, 17, 35] = 29, μ3 [11, 17, 23] = 17, μ3 [11, 17, 29] = 23, μ3 [11, 17, 35] = 29, μ3 [11, 23, 23] = 23, μ3 [11, 23, 29] = 29, μ3 [11, 23, 23] = 23, μ3 [11, 23, 35] = 35, μ3 [11, 29, 29] = 35, μ3 [11, 29, 35] = 5, μ3 [11, 35, 35] = 11, μ3 [17, 17, 17] = 17, μ3 [17, 17, 23] = 23, μ3 [17, 17, 29] = 29, μ3 [17, 17, 35] = 35, μ3 [17, 23, 23] = 29, μ3 [17, 23, 29] = 35, μ3 [17, 29, 29] = 5, μ3 [17, 29, 35] = 11, μ3 [17, 35, 35] = 17, μ3 [23, 23, 23] = 35, μ3 [23, 23, 29] = 5, μ3 [23, 23, 35] = 11, μ3 [23, 29, 29] = 11, μ3 [23, 29, 35] = 17, μ3 [23, 35, 35] = 23, μ3 [29, 29, 29] = 17, μ3 [29, 29, 35] = 23, μ3 [29, 35, 35] = 29, μ3 [35, 35, 35] = 35.

The multiplicative querelements are ˉ 5

= 29, 29 = 5, 11 = 23, 23 = 11.

137

18 Multiplicative properties of exotic finite polyadic fields

Because

5h1i×3 = 17e , 5h2i×3 = 29, 5h3i×3 = 5, 29h1i×3 = 17e ,

(18.1)

29h2i×3 = 5, 29h3i×3 = 29,

(18.2)

11h1i×3 = 35e , 11h2i×3 = 23, 11h3i×3 = 11, 23h1i×3 = 35e ,

(18.3)

23h2i×3 = 11, 23h3i×3 = 23,

(18.4)

[5,6]

the multiplicative 3-ary group G(7,3) (6) consists of two nonintersecting cyclic

3-ary subgroups

[5,6]

G(7,3) (6) = G1 ∪ G2 , G1 ∩ G2 = ∅, G1 = {5, 17e , 29} ,

G2 = {11, 23, 35e } ,

(18.5) (18.6) (18.7)

which is impossible for binary subgroups, as these always intersect in the identity of a binary group.

138

18 Multiplicative properties of exotic finite polyadic fields

[5,6]

2) The finite polyadic field F(7,3) (4)

= {{5, 11, 17, 23} | ν 7 , μ3 } which has the same arity shape as above, but with order 4, has the exotic property that all

elements are units, which follows from its ternary multiplication table

μ3 [5, 5, 5] = 5, μ3 [5, 5, 11] = 11, μ3 [5, 5, 17] = 17, μ3 [5, 5, 23] = 23, μ3 [5, 11, 11] = 5, μ3 [5, 11, 17] = 23, μ3 [5, 11, 23] = 17, μ3 [5, 17, 17] = 5, μ3 [5, 17, 23] = 11, μ3 [5, 23, 23] = 5, μ3 [11, 11, 11] = 11, μ3 [11, 11, 17] = 17, μ3 [11, 11, 23] = 23, μ3 [11, 17, 17] = 11, μ3 [11, 17, 23] = 5, μ3 [11, 23, 23] = 11, μ3 [17, 17, 17] = 17, μ3 [17, 17, 23] = 23, μ3 [17, 23, 23] = 17, μ3 [23, 23, 23] = 23.

139

18 Multiplicative properties of exotic finite polyadic fields

3) Next we show by construction, that (as opposed to the case of binary finite fields) there exist non-isomorphic finite polyadic fields of the same order and arity [3,8]

shape. Indeed, consider these two (9, 3)-fields of order 2, that are F(9,3) (2) [7,8]

and F(9,3) (2). The first is zeroless-nonunital, while the second is zeroless with [3,8]

two units, i.e. all elements are units. The multiplication of F(9,3) (2) is

μ3 [3, 3, 3] = 11, μ3 [3, 3, 11] = 3, μ3 [3, 11, 11] = 11, μ3 [11, 11, 11] = 3, 3 having the multiplicative querelements ˉ

[7,8]

= 11, 11 = 3. For F(9,3) (2) we get

the 3-group of units

μ3 [7, 7, 7] = 7, μ3 [7, 7, 15] = 15, μ3 [7, 15, 15] = 7, μ3 [15, 15, 15] = 15. [3,8]

They have different idempotence polyadic orders ord F(9,3) (2)

= 2 and

[7,8]

ord F(9,3) (2) = 1. Despite their additive m-ary groups being isomorphic, it follows from the above multiplicative structure, that it is not possible to construct an isomorphism between the fields themselves.

140

18 Multiplicative properties of exotic finite polyadic fields

4) The smallest exotic finite polyadic field with more than one unit is [2,3]

F(4,3) (5) = {{2, 5, 8, 11, 14} | ν 4 , μ3 } of order 5 with two units {11, 14} ≡ {11e , 14e } and the zero 5 ≡ 5z . The additive querelements are ˜2 = 11e , ˜ 8 = 14e , f 11e = 8, f 14e = 2.

(18.8)

2h2i×3 = 2, 8h2i×3 = 8,

(18.9)

[2,3]

The idempotence polyadic order is ord F(4,3) (5)

= 2, because

and their multiplicative querelements are ˉ 2

= 8, ˉ 8 = 2. The multiplication is [2,3] (4) = {{2, 8, 11, 14} | μ3 } as given by the cyclic 3-ary group G3 μ3 [2, 2, 2] = 8, μ3 [2, 2, 8] = 2, μ3 [2, 2, 11] = 14, μ3 [2, 2, 14] = 11, μ3 [2, 8, 8] = 8, μ3 [2, 8, 11] = 11, μ3 [2, 8, 14] = 14, μ3 [2, 11, 11] = 2, μ3 [2, 11, 14] = 8, μ3 [2, 14, 14] = 2, μ3 [8, 8, 8] = 2, μ3 [8, 8, 11] = 14, μ3 [8, 8, 14] = 11, μ3 [8, 11, 11] = 8, μ3 [8, 11, 14] = 2, μ3 [8, 14, 14] = 8, μ3 [11, 11, 11] = 11, μ3 [11, 11, 14] = 14, μ3 [11, 14, 14] = 11, μ3 [14, 14, 14] = 14.

141

18 Multiplicative properties of exotic finite polyadic fields

[2,3]

In spite of the fact that the cyclic 3-ary group G3

(4) has two units, no

decomposition into nonintersecting cyclic 3-ary subgroups, as in (18.5). [5,6]

One of the reasons is that the polyadic field F(7,3) (6) is zeroless, while [2,3]

F(4,3) (5) has a zero. Conclusion Recall that any binary finite field has an order which is a power of a prime number

q = pr (its characteristic), and all such fields are isomorphic and contain a prime subfield GF (p) of order p which is isomorphic to the congruence (residue) class field ZpZ L IDL AND N IEDERREITER [1997]. Conjecture 18.1. A finite (m, n)-field (with m > n) should contain a minimal subfield which is isomorphic to one of the prime polyadic fields constructed [a,b]

above, and therefore the introduced here finite polyadic fields F(m,n) (q) can be interpreted as polyadic analogs of the prime Galois field GF

142

(p).

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

19 Equal sums of like powers Diophantine equation over polyadic integer numbers Recall the standard binary version of the equal sums of like powers Diophantine equation L ANDER

ET AL .

[1967], E KL [1998] p+1 X i=1

ul+1 = i

q+1 X

vjl+1 = r,

(19.1)

j=1

∈ N0 , p ≤ q , and the positive integer unknowns placed in ascending order ui ≤ ui+1 , vj ≤ vj+1 , ui , vj ∈ Z+ , i = 1, . . . p + 1, j = 1, 1, . . . q + 1. We mark the solutions of (19.1) by the triple (l | p, q)r where p, q, l

showing quantity of operations. In the binary case, the solutions of (19.1) are

usually denoted by (l + 1

| p + 1, q + 1)r , which shows the number of

summands on both sides and powers of elements L ANDER

ET AL .

[1967]. But in

the polyadic case, the number of summands and powers do not coincide with

l + 1, p + 1, q + 1, at all. Here r (if it is used) is the order of the solution. 143

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Let us recall the Tarry-Escott problem (or multigrades problem) D ORWART

AND

B ROWN [1937]: to find the solutions to (19.1) for an equal number of summands on both sides of p

= q and s equations simultaneously, such that l = 0, . . . , s. Known solutions exist for powers until s = 10, which are bounded such that s ≤ p (in our notations), see, also, N GUYEN [2016]. The solutions with highest powers s = p are called the ideal solutions B ORWEIN [2002]. Theorem 19.1 (Frolov F ROLOV [1889]). If the set of s Diophantine equations ( 19.1) with p = q for l = 0, . . . , s has a solution {ui , vi , i = 1, . . . p + 1}, then it has the solution {a + bui , a + bvi , i = 1, . . . p + 1}, where a, b ∈ Z. In the simplest case (1

| 0, 1), one term in l.h.s., one addition on the r.h.s. and

one multiplication, the (coprime) positive numbers satisfying (19.1) are called a (primitive) Pythagorean triple. For the Fermat’s triple (l on the r.h.s. and more than one multiplication l

| 0, 1) with one addition

≥ 2, there are no solutions of

(19.1) , which is known as Fermat’s last theorem proved in W ILES [1995]. There are many solutions known with more than one addition on both sides, where the highest number of multiplications till now is 31 (S. Chase, 2012).

144

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Before generalizing (19.1) for polyadic case we note the following. Remark 19.2. The notations in (19.1) are chosen in such a way that p and q are numbers of binary additions on both sides, while l is the number of binary multiplications in each term, which is consistent with the usage of polyadic powers D UPLIJ [2012].

Polyadic analog of the Lander-Parkin-Selfridge conjecture In L ANDER

ET AL .

[1967], a generalization of Fermat’s last theorem was

conjectured, that the solutions of (19.1) exist for small powers only. Conjecture 19.3 (Lander-Parkin-Selfridge L ANDER

ET AL .

[1967]). There exist

solutions of (19.1) in positive integers, if the number of multiplications is less than or equal than the total number of additions plus one

3 ≤ l ≤ lLSP = p + q + 1, where p + q

≥ 2. 145

(19.2)

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Remark 19.4. If the equation (19.1) is considered over the binary ring of integers

Z, such that ui , vj ∈ Z, it leads to a straightforward reformulation: for even

powers it is obvious, but for odd powers all negative terms can be rearranged and placed on the other side. Let us consider the Diophantine equation (19.1) over polyadic integers Z(m,n)

(i.e. over the polyadic (m, n)-ary ring RZ m,n ) such that ui , vj (l)

(l)

∈ RZm,n . We use

the “long products” μn and ν m containing l operations, and also the “polyadic

power” for an element x

In the binary case, n element xhli2

∈ RZm,n D UPLIJ [2012]   l(n−1)+1 }| { hlin (l) z x = μn x, x, . . . , x .

(19.3)

= 2, the polyadic power coincides with (l + 1) power of an

= xl+1 , which explains R EMARK 19.2.

146

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

The polyadic analog of the equal sums of like powers Diophantine equation is

h

hlin hlin hlin u ν (p) , u , . . . , u m 1 2 p(m−1)+1

i

=

h

hlin hlin hlin ν (q) v , v , . . . , v m 1 2 q(m−1)+1

i

,

(19.4)

where p and q are number of m-ary additions in l.h.s. and r.h.s. correspondingly. The solutions of (19.4) will be denoted by





u1 , u2 , . . . , up(m−1)+1 ; v1 , v2 , . . . , vq(m−1)+1 .

In the binary case m

(19.5)

= 2, n = 2, (19.4) reduces to (19.1). Analogously, we (m,n) . mark the solutions of (19.4) by the polyadic triple (l | p, q)r (m,n)

, having one term on the l.h.s., | 0, 1) one m-ary addition on the r.h.s. and one n-ary multiplication (elements are in the first polyadic power h1in ), becomes h i h1in h1in h1in h1in (19.6) u1 = ν m v 1 , v 2 , . . . , v m . The polyadic Pythagorean triple (1

Definition 19.5. The equation (19.6) solved by minimal u1 , vi

∈ Z,

i = 1, . . . , m can be named the polyadic Pythagorean theorem. 147

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

The polyadic Fermat’s triple (l

| 0, 1)

(m,n)

has one term in l.h.s., one m-ary

addition in r.h.s. and l (n-ary) multiplications hli u1 n

=

h

hli hli hli ν m v1 n , v 2 n , . . . , v m n

i

.

(19.7)

Conjecture 19.6 (Polyadic analog of Fermat’s Last Theorem). The polyadic Fermat’s triple (19.7) has no solutions over the polyadic (m, n)-ary ring RZ m,n , if

l ≥ 2, i.e. there are more than one n-ary multiplications.

Its straightforward generalization leads to the polyadic version of the Lander-Parkin-Selfridge conjecture, as Conjecture 19.7 (Polyadic Lander-Parkin-Selfridge conjecture). There exist solutions of the polyadic analog of the equal sums of like powers Diophantine equation (19.4) in integers, if the number of n-ary multiplications is less than or equal than the total number of m-ary additions plus one

3 ≤ l ≤ lpLP S = p + q + 1. Below we will see counterexamples to both of the above conjectures.

148

(19.8)

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Example 19.8. Let us consider the (3, 2)-ring RZ 3,2

= hZ | ν 3 , μ2 i, where

ν 3 [x, y, z] = x + y + z + 2,

(19.9)

μ2 [x, y] = xy + x + y.

(19.10)

Note that this exotic polyadic ring is commutative and cancellative, having unit 0, no multiplicative inverses, and for any x

∈ RZ3,2 its additive querelement

x ˜ = −x − 2, therefore hZ | ν 3 i is a ternary group (as it should be). The polyadic power of any element is

xhli2 = (x + 1)

l+1

1) For RZ 3,2 the polyadic Pythagorean triple (1

h

− 1.

| 0, 1)

(19.11)

(3,2)

i

uh1i2 = ν 3 xh1i2 , y h1i2 , z h1i2 .

149

in ( 19.6) now is (19.12)

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

This equation becomes the (shifted) Pythagorean quadruple S PIRA [1962] 2

2

2

2

(u + 1) = (x + 1) + (y + 1) + (z + 1) ,

(19.13)

and it has infinite number of solutions, among which two minimal ones

{u = 2; x = 0, y = z = 1} and {u = 14; x = 1, y = 9, z = 10} give 32 = 12 + 22 + 22 and 152 = 22 + 102 + 112 , correspondingly. 2) For this (3, 2)-ring RZ 3,2 the polyadic Fermat’s triple (l

(u + 1)

l+1

= (x + 1)

l+1

+ (y + 1)

l+1

| 0, 1)

+ (z + 1)

(3,2)

l+1

.

becomes (19.14)

If the polyadic analog of Fermat’s last theorem 19.6 holds, then there are no solutions to ( 19.14) for more than one n-ary multiplication l particular case, p

≥ 2. But this is the

= 0, q = 2, of the binary Lander-Parkin-Selfridge Conjecture 19.3 which now takes the form: the solutions to ( 19.14) exist, if l ≤ 3. Thus, as a counterexample to the polyadic analog of Fermat’s last theorem, we have two

possible solutions with numbers of multiplications:

150

l = 2, 3.

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

In the case of l

= 2 there exist two solutions: one well-known solution {u = 5; x = 2, y = 3, z = 4} giving 63 = 33 + 43 + 53 and another one giving 7093 = 1933 + 4613 + 6313 (J.-C. Meyrignac, 2000), while for l = 3 there exist an infinite number of solutions, and one of them (minimal) gives

4224814 = 958004 + 2175194 + 4145604 E LKIES [1988]. 3) The general polyadic triple (l

| p, q)

(3,2)

, using ( 19.4), can be presented in

the standard binary form (as ( 19.1)) 2p+1 X i=1

(ui + 1)

l+1

=

2q+1 X

(vi + 1)

j=1

l+1

, ui , vj ∈ Z.

(19.15)

Let us apply the polyadic Lander-Parkin-Selfridge Conjecture 19.7 for this case: the solutions to ( 19.15) exist, if 3

≤ l ≤ lpLSP = p + q + 1. But the binary

Lander-Parkin-Selfridge Conjecture 19.3, applied directly, gives

3 ≤ l ≤ lLSP = 2p + 2q + 1. So we should have counterexamples to the polyadic Lander-Parkin-Selfridge conjecture, when lpLSP < l ≤ lLSP . 151

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

For instance, for p

= q = 1, we have lpLSP = 3, while the (minimal) counterexample with l = 5 is {u1 = 3, u2 = 18, u3 = 21, v1 = 9, v2 = 14, v3 = 22} giving 36 + 196 + 226 = 106 + 156 + 236 S UBBA R AO [1934]. As it can be observed from Example 19.8, the arity shape of the polyadic ring

RZm,n is crucial in constructing polyadic analogs of the equal sums of like powers conjectures. We can make some general estimations assuming a special (more or less natural) form of its operations over integers. Definition 19.9. We call RZ m,n the standard polyadic ring, if the “leading terms” of its m-ary addition and n-ary multiplication are

νm

"

m

#

"

z }| { x, x, . . . , x ∼ mx, n

#

z }| { μn x, x, . . . , x ∼ xn , x ∈ Z. 152

(19.16)

(19.17)

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

The polyadic ring RZ 3,2 from Example 19.8, as well as the congruence class [a,b]

polyadic ring Rm,n are both standard.

Using (19.3), we obtain approximate behavior of the polyadic power in the standard polyadic ring

xhlin ∼ xl(n−1)+1 , x ∈ Z, l ∈ N, n ≥ 2.

(19.18)

So increasing the arity of multiplication leads to higher powers, while increasing arity of addition gives more terms in sums. Thus, the estimation for the polyadic analog of the equal sums of like powers Diophantine equation (19.4) becomes

(p (m − 1) + 1) xl(n−1)+1 ∼ (q (m − 1) + 1) xl(n−1)+1 , x ∈ Z.

(19.19)

Now we can apply the binary Lander-Parkin-Selfridge Conjecture 19.3 in the form: the solutions to (19.19) can exist if 3 integer solution of

≤ l ≤ lLP S , where lLP S is an

(n − 1) lLP S = (p + q) (m − 1) + 1. 153

(19.20)

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

On the other hand the polyadic Lander-Parkin-Selfridge Conjecture 19.7 gives: the solutions to (19.19) can exist if 3

(p + q) ≥ 2 now.

≤ l ≤ lpLP S = p + q + 1. Note that

An interesting question arises: which arities give the same limit, that is, when

lpLP S = lLP S ? Proposition 19.10. For any fixed number of additions in both sizes of the polyadic analog of the equal sums of like powers Diophantine equation ( 19.4)

p + q ≥ 2, there exist limiting arities m0 and n0 (excluding the trivial binary case m0 = n0 = 2), for which the binary and polyadic Lander-Parkin-Selfridge conjectures coincide lpLP S = lLP S , such that m0 = 3 + p + q + (p + q + 1) k, n0 = 2 + p + q + (p + q) k, k ∈ N0 .

154

(19.21) (19.22)

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Corollary 19.11. In the limiting case lpLP S

= lLP S the arity of multiplication

always exceeds the arity of addition

m0 − n0 = k + 1, k ∈ N0 , and they start from m0

(19.23)

≥ 5, n0 ≥ 4.

The first allowed arities m0 and n0 are presented in TABLE 4. Their meaning is the following. Corollary 19.12. For the polyadic analog of the equal sums of like powers equation over the standard polyadic ring RZ m,n (with fixed p + q

≥ 2) the

polyadic Lander-Parkin-Selfridge conjecture becomes weaker than the binary one

lpLP S ≥ lLP S , if: 1) the arity of multiplication exceeds its limiting value n0 with fixed arity of the addition; 2) the arity of addition is lower, than its limiting value m0 with the fixed arity of multiplication.

155

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Table 4: The limiting arities m0 and n0 , which give lpLP S

= lLP S in (19.19).

p+q =2

p+q =3

p+q =4

m0

n0

m0

n0

m0

n0

5

4

6

5

7

6

8

6

10

8

12

10

11

8

14

11

17

14

14

10

18

14

22

18

156

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Example 19.13. Consider the standard polyadic ring RZ m,n and fix the arity of addition m0

= 12, then take in ( 19.19) the total number of additions p + q = 4 (the last column in TABLE 3). The arity of multiplication n = 16, which exceeds the limiting arity n0 = 10 (corresponding to m0 ). Thus, we obtain lpLP S = 5 and lLP S = 3 by solving ( 19.20) in integers, and therefore the polyadic Lander-Parkin-Selfridge conjecture becomes now weaker than the binary one, and we do not obtain counterexamples to it, as in Example 19.8 (where the situation was opposite lpLP S

= 3 and lLP S = 5, and they cannot be equal).

A concrete example of the standard polyadic ring (Definition 19.9) is the polyadic [a,b]

ring of the fixed congruence class Rm,n considered before, because its

operations (17.5)–(17.6) have the same straightforward behavior (19.16)–(19.17).

157

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Let us formulate the polyadic analog of the equal sums of like powers Diophantine [a,b]

equation (19.4) over Rm,n in terms of operations in Z. Using (17.5)–(17.6) and

(19.18) for (19.4) we obtain p(m−1)+1

X

q(m−1)+1

(a + bki )

l(n−1)+1

=

i=1

X j=1

(a + bkj )

l(n−1)+1

, a, b, ki ∈ Z. (19.24)

It is seen that the leading power behavior of both sides in (19.24) coincides with the general estimation (19.19). But now the arity shape (m, n) is fixed by (17.7)–(17.8) and given in TABLE 2. Nevertheless, we can consider for (19.24) the polyadic analog of Fermat’s last theorem 19.6, the Lander-Parkin-Selfridge

≤ lLP S ) and its polyadic version (Conjecture 19.7, solutions exist for l ≤ lpLP S ), as in the estimations above.

Conjecture 19.3 (solutions exist for l

158

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Let us consider some examples of solutions to (19.24). Example 19.14. Let [[2]]3 be the congruence class, which is described by [2,3]

(4, 3)-ring R4,3 (see TABLE 1), and we consider the polyadic Fermat’s triple (l | 0, 5)

(4,3)

( 19.7). Now the powers are lLP S

= 8, lpLP S = 6, and for instance, if l = 2, we have solutions, because l < lpLP S < lLP S , and one of

them is 5

145 = 4 ∙ (−1) + 7 ∙ 55 + 85 + 2 ∙ 115 .

(19.25)

Frolov’s Theorem and the Tarry-Escott problem A special set of solutions to the polyadic Lander-Parkin-Selfridge Conjecture 19.7 can be generated, if we put p

= q in (19.24), which we call equal-summand

solutionsa , by exploiting the Tarry-Escott problem approach D ORWART B ROWN [1937] and Frolov’s Theorem 19.1. a

The term “symmetric solution” is already taken and widely used B ORWEIN [2002].

159

AND

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Theorem 19.15. If the set of integers ki p(m−1)+1

X i=1

∈ Z solves the Tarry-Escott problem

p(m−1)+1

kir

=

X

kjr , r = 1, . . . , s = l (n − 1) + 1,

j=1

(19.26)

then the polyadic equal sums of like powers equation with equal summands [a,b]

( 19.4) has a solution over the polyadic (m, n)-ring Rm,n having the arity shape given by the following relations: 1. Inequality

l (n − 1) + 1 ≤ p (m − 1) ;

(19.27)

p (m − 1) = 2l(n−1)+1 .

(19.28)

2. Equality

160

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Let us consider some examples which give solutions to the polyadic equal sums of like powers equation (19.4) with p the fixed congruence class [[a]]b .

[a,b]

= q over the polyadic (m, n)-ring Rm,n of

Example 19.16. 1) One of the first ideal (non-symmetric) solutions to the Tarry-Escott problem has 6 summands and 5 powers (A. Golden, 1944)

0r +19r +25r +57r +62r +86r = 2r +11r +40r +42r +69r +85r , r = 1, . . . , 5. (19.29) We compare with ( 19.26) and obtain

p (m − 1) = 5, l (n − 1) = 4.

After ignoring binary arities we get m

(19.30) (19.31)

= 6, p = 1 and n = 3, l = 2. From TABLE 1 we observe the minimal choice a = 4 and b = 5. It follows from Frolov’s

Theorem 19.1, that all equations in ( 19.29) have symmetry

ki → a + bki = 4 + 5ki . 161

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Thus, we obtain the solution of the polyadic equal sums of like powers equation ( 19.4) for the fixed congruence class [[4]]5 in the form

45 +995 +1295 +2895 +3145 +4345 = 145 +595 +2045 +2145 +3495 +4295 . (19.32) It is seen from TABLE 1 that the arity shape (m

= 6, n = 3) corresponds, e.g.,

to the congruence class [[4]]10 as well. Using Frolov’s theorem, we substitute in ( 19.29) ki

→ 4 + 10ki to obtain the solution in the congruence class [[4]]10

45 +1945 +2545 +5745 +6245 +8645 = 245 +1145 +4045 +4245 +6945 +8545 . (19.33) 2) To obtain the special kind of solutions to the Tarry-Escott problem we start with the known one with 8 summands and 3 powers (see, e.g., L EHMER [1947])

0r + 3r + 5r + 6r + 9r + 10r + 12r + 15r = 1r + 2r + 4r + 7r + 8r + 11r + 13r + 14r , r = 1, 2, 3.

162

(19.34)

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

So we have the concrete solution to the system ( 19.26) with the condition ( 19.28) which now takes the form 8

= 23 , and therefore p (m − 1) = 7,

(19.35)

l (n − 1) = 2.

Excluding the trivial case containing binary arities, we have m

(19.36)

= 8, p = 1 and

n = 3, l = 1. It follows from TABLE 1, that a = 6 and b = 7, and so the [6,7] polyadic ring is R8,3 . Using Frolov’s Theorem 19.1, we can substitute entries in ( 19.34) as ki → a + bki = 6 + 7ki in the equation with highest power r = 3 (which is relevant to our task) and obtain the solution of ( 19.4) for [[6]]7 as follows 63 + 273 + 413 + 483 + 693 + 763 + 903 + 1113 = 133 + 203 + 343 + 553 + 623 + 833 + 973 + 1043 .

163

(19.37)

19 Equal sums of like powers Diophantine equation over polyadic integer numbers

Conclusion Consideration of the Tarry-Escott problem and Frolov’s theorem over polyadic rings gives the possibility of obtaining many nontrivial solutions to the polyadic equal sums of like powers equation for fixed congruence classes.

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