Steven Duplij, "Developing new supermanifolds by revitalizing old ideas"

Developing new supermanifolds
by revitalizing old ideas
STEVEN DUPLIJ
M¨unster
http://wwwmath.uni-muenster.de/u/duplij
Luxemburg-2017
1

Plan
1. Regular mappings and semisupermanifolds
2. Superconformal-like transformations twisting parity of tangent space
3. Ternary supersymmetry
4. Polyadic analogs of integer number ring Z and field Z/pZ
2

1 Regular mappings and semi-supermanifolds
The concept of regularity was introduced by von Neumann VON NEUMANN [1936]
and Penrose for matrices PENROSE [1955].
Von Neumann regularity. Let R be a ring. If for an element a ∈ R there is an
element a such that
aa a = a, a aa = a , (1.1)
then a is said to be regular and a is a generalized inverse of a. Study of such
regularity and related directions was developed in generalized inverses theory
RABSON [1969], NASHED [1976], semigroup theory HOWIE [1976], LAWSON
[1998], supermanifold theory DUPLIJ [2000], weak bialgebras and Hopf algebras
NILL [1998], category theory DAVIS AND ROBINSON [1972].
3

Noninvertibility in supermanifolds: Motivations
• The Hopf spaces in general have no conditions on existence of inverses (in
homotopy sense) STASHEFF [1970]
• “...a general SRS needs not have a body” CRANE AND RABIN [1988] and “...a
body may not even exist” BRYANT [1988]
• “...there may be no inverse projection (body map ROGERS [1980]) at all”
PENKOV [1981]
• “The interpretation of anticommuting variables can be also dramatically
changed in future” YAPPA [1987]
• There exist (topologically) quasinilpotent odd elements which are not nilpotent
PESTOV [1991, 1992]
• The possibility of definition of a supermanifold without the notion of
topological space MINACHIN [1988].
4

Standard patch definition of a supermanifold
Recall the (informal) patch definition of a supermanifold RABIN AND CRANE
[1985a], ROGERS [1980], VLADIMIROV AND VOLOVICH [1984].
Consider a collection of superdomains Uα such that a supermanifold is
M0 =
α
Uα. For Uα we take some superfunctions (coordinate maps)
ϕα : Uα → Dn|m
⊂ Rn|m
, where Rn|m
is a superspace in which our super
“ball” lives, and Dn|m
is an open domain in Rn|m
.
The pair {Uα, ϕα} is a local chart and the union of charts
α
{Uα, ϕα} is an
atlas of a supermanifold.
5

Introduce gluing transition functions on Uαβ = Uα ∩ Uβ = ∅ by
Ψαβ = ϕα ◦ ϕ−1
β (1.2)
The transition functions Ψαβ give us possibility to restore the whole
supermanifold from individual charts and coordinate maps. The transition
functions satisfy the
Ψ−1
αβ = Ψβα (1.3)
on Uα ∩ Uβ and
Ψαβ ◦ Ψβγ ◦ Ψγα = 1αα (1.4)
on triple overlaps Uα ∩ Uβ ∩ Uγ, where 1αα
def
= id (Uα). The maps ϕα are
homeomorphisms, no distinguish between Uα and Dn|m
, i.e. locally
supermanifolds are as the whole superspace Rn|m
.
6

Von Neumann and n-regular extension of supermanifold
We formulate a patch definition of an object analogous to a supermanifold, i.e. try
to weaken demand of invertibility of ϕα as maps. Let us consider a generalized
superspace M covered by open sets Uα as M =
α
Uα. We assume here that
the maps ϕα are not all homeomorphisms.
Definition 1.1. A chart is a pair Uinv
α , ϕinv
α ,where ϕinv
α is an invertible
morphism. A semichart is a pair Unoninv
α , ϕnoninv
α , where ϕnoninv
α is a
noninvertible morphism.
7

Definition 1.2. A semiatlas {Uα, ϕα} is a union of charts and semicharts
Uinv
α , ϕinv
α Unoninv
α , ϕnoninv
α . (1.5)
Definition 1.3. A semisupermanifold is a noninvertibly generalized superspace M
represented as a semiatlas M =
α
{Uα, ϕα}.
Definition 1.4. The gluing noninvertible semitransition functions of a
semisupermanifold are defined by the equations
Φαβ ◦ ϕβ = ϕα, Φβα ◦ ϕα = ϕβ. (1.6)
Instead of (1.3) Φαβ ◦ Φβα = 1αα we have
Conjecture 1.5. The semitransition functions Φαβ of a semisupermanifold, on
Uα ∩ Uβ overlaps, satisfy
Φαβ ◦ Φβα ◦ Φαβ = Φαβ. (1.7)
8

Instead of (1.4) Φαβ ◦ Φβγ ◦ Φγα = 1αα we propose
Φαβ ◦ Φβγ ◦ Φγα ◦ Φαβ = Φαβ, (1.8)
Φβγ ◦ Φγα ◦ Φαβ ◦ Φβγ = Φβγ, (1.9)
Φγα ◦ Φαβ ◦ Φβγ ◦ Φγα = Φγα (1.10)
on triple overlaps Uα ∩ Uβ ∩ Uγ and
Φαβ ◦ Φβγ ◦ Φγρ ◦ Φρα ◦ Φαβ = Φαβ, (1.11)
Φβγ ◦ Φγρ ◦ Φρα ◦ Φαβ ◦ Φβγ = Φβγ, (1.12)
Φγρ ◦ Φρα ◦ Φαβ ◦ Φβγ ◦ Φγρ = Φγρ, (1.13)
Φρα ◦ Φαβ ◦ Φβγ ◦ Φγρ ◦ Φρα = Φρα (1.14)
on Uα ∩ Uβ ∩ Uγ ∩ Uρ, etc.
Definition 1.6. We call (1.7)–(1.14) n-regular tower relations.
The functions Φαβ can be viewed as some noninvertible generalization of
cocycles in the ˇCech cohomology of coverings MACLANE [1967].
9

In supersymmetry the role of Jacobian plays Berezinian BEREZIN [1987] which
has a “sign” belonging to Z2 ⊕ Z2 VORONOV [1991], TUYNMAN [1995], and so
there are four orientations on Uα and five corresponding kinds of orientability
SHANDER [1988].
Definition 1.7. In case a nonzero Berezinian of Φαβ is nilpotent (and so has no
definite sign) there exists additional nilpotent orientation on Uα of a
semisupermanifold.
A degree of nilpotency of the Berezinian allows us to classify semisupermanifolds
having nilpotent orientability.
10

Semisupermanifolds and obstruction
The semisupermanifolds defined above belong to a class of so called obstructed
semisupermanifolds in the following sense. Let us rewrite (1.2), (1.3) and (1.4) as
the infinite series
n = 1 : Φαα = 1αα, (1.15)
n = 2 : Φαβ ◦ Φβα = 1αα, (1.16)
n = 3 : Φαβ ◦ Φβγ ◦ Φγα = 1αα, (1.17)
n = 4 : Φαβ ◦ Φβγ ◦ Φγδ ◦ Φδα = 1αα, . . . (1.18)
Definition 1.8. A semisupermanifold is called obstructed if some of the cocycle
conditions (1.15)–(1.18) are broken.
Definition 1.9. The obstruction degree of a semisupermanifold is such nm for
which the cocycle conditions (1.15)–(1.18) are broken.
If all (1.15)–(1.18) hold valid, then we say that nm
def
= ∞.
11

We can compare semisupermanifolds with supernumbers:
Supernumbers Semisupermanifolds
Ordinary nonzero numbers (invertible)
Ordinary manifolds (transition functions
are invertible)
Supernumbers having a nonvanishing
body part (invertible)
Supermanifolds (transition functions
are invertible)
Pure soul supernumbers without a
body part (noninvertible)
Obstructed semisupermanifolds
(transition functions are noninvertible)
12

Let us consider a series of the selfmaps e
(n)
αα : Uα → Uα of a semi-manifold
defined as
e(1)
αα = Φαα, (1.19)
e(2)
αα = Φαβ ◦ Φβα, (1.20)
e(3)
αα = Φαβ ◦ Φβγ ◦ Φγα, (1.21)
e(4)
αα = Φαβ ◦ Φβγ ◦ Φγδ ◦ Φδα (1.22)
∙ ∙ ∙ ∙ ∙ ∙
We call the maps e
(n)
αα tower identities. For an ordinary supermanifolds all tower
identities coincide with the identity map (id)
e(n)
αα = 1αα. (1.23)
The obstruction degree can be treated as such n = nm for which the tower
identities (1.19)–(1.22) differ from the identity.
13

The tower identities are left units for the semitransition functions
e(n)
αα ◦ Φαβ = Φαβ. (1.24)
It is obvious, that the tower identities are idempotents
e(n)
αα ◦ e(n)
αα = e(n)
αα . (1.25)
Definition 1.10. A semisupermanifold is nice, if the tower identities e
(n)
αα do not
depend on a given partition.
The multiplication of the tower identities of a nice semi-supermanifold can be
defined as follows
e(n)
αα ◦ e(m)
αα = e(n+m)
αα . (1.26)
The multiplication (1.26) is associative, and therefore
Corollary 1.11. The tower identities of a nice semisupermanifold form a tower
semigroup under the multiplication (1.26).
14

The extension of n = 2 cocycle (1.7) Φαβ ◦ Φβα ◦ Φαβ = Φαβ can be viewed
as some analogy with regular elements in semigroups CLIFFORD AND PRESTON
[1961], PETRICH [1984] or generalized inverses in matrix theory PENROSE
[1955], RAO AND MITRA [1971], category theory DAVIS AND ROBINSON [1972]
and theory of generalized inverses of morphisms NASHED [1976].
Definition 1.12. Noninvertible mappings Φαβ are n-regular, if they satisfy
n
Φαβ ◦ Φβγ ◦ . . . ◦ Φρα ◦ Φαβ= Φαβ (1.27)
+ perm
on overlaps
n
Uα ∩ Uβ ∩ . . . ∩ Uρ.
15

Summarizing the above statements, we propose the following regularization of the
standard diagram technique
Invertible
Φαβ
Φβα
=⇒
Noninvertible
Φβα
Φαβ
n = 2
Φαβ
Φγα
=⇒ + perm
Φβγ Φγα
Φαβ
Φβγ
n = 3
16

Example 1.13. Every inverse semigroup S with elements satisfying xyx = x
and yxy = y is a nontrivial 2-regular obstructed category. It has only one object,
morphisms are the elements of S.
Example 1.14 ( n-regular analog of the Grassmann algebra DUPLIJ AND
MARCINEK [2001]). Let us consider an associative algebra A = Λ (Θ, Θ∗
)
generated by two noncommuting generators Θ and Θ∗
satisfying
Θ2
= Θ∗2
= 0, ΘΘ∗
Θ = Θ, Θ∗
ΘΘ∗
= Θ∗
. (1.28)
Denote by |a| the parity of a ∈ A, by |ab| = |a| + |b| (mod2) for the
multiplication in A. Let a, b ∈ A be odd elements, |a| = |b| = 1, then the
product ab is an even element of A, |ab| = 0(mod2). We take the generators
Θ and Θ∗
to be odd, then A = Λ (Θ, Θ∗
) is 2-regular analog of the
Grassmann algebra.
The algebra A = Λ (Θ, Θ∗
) is free product of two one dimensional Grassmann
algebras Λ(Θ) and Λ(Θ∗
) modulo relations (1.28).
17

Example 1.15. Let us consider n copies of the one–dimensional Grassmann
algebra Λ (Θ). The i-th copy is denoted by Λ(Θ
i
∗ ∗ ...∗). Let us define a
superalgebra Λ(Θ, Θ∗
, . . . , Θ
n−1
∗ ∗ ...∗) as a free product of n copies of
one–dimensional Grassmann algebras subject to the following relation
ΘΘ∗
Θ∗∗
∙ ∙ ∙ Θ
n−1
∗ ∗ ...∗ Θ = Θ (1.29)
and its all cyclic permutations.
Definition 1.16. The algebra Λ(Θ, Θ∗
, . . . , Θ
n−1
∗ ∗ ...∗) is n–regular analog of
the Grassmann algebra DUPLIJ AND MARCINEK [2001].
18

Example 1.17 (Braid semistatistics DUPLIJ AND MARCINEK [2003]). The
standard statistics and its generalizations GREENBERG AND SERGIESCU [1991],
MARCINEK [1991] are “invertible” in the following sense.
12
a
→ 21
b
→ 12 = 12
id
→ 12 “invertibility”, (1.30)
Weaken it to the Von Neumann regularity by
12
a
→ 21
b
→ 12
a
→ 21 = 12
a
→ 21 “left regularity”, (1.31)
21
b
→ 12
a
→ 21
b
→ 12 = 21
b
→ 12 “right regularity”. (1.32)
So the usual statistics as one morphism a, because b can be found from the
“invertibility” condition (1.30) which is a ◦ b = id symbolically).
We introduce the more abstract concept of “semistatistics” as a pair of
exchanging morphisms a and b satisfying the “regularity” conditions (1.31)–(1.32)
(symbolically a ◦ b ◦ a = a, b ◦ a ◦ b = b) . Then we introduce the von
Neumann regular analogs of braidings and YBE DUPLIJ AND MARCINEK [2003].
19

2 Superconformal-like transformations twisting parity of tangent space
2 Superconformal-like transformations twisting parity of
tangent space
Super Riemann surfaces are particular case of (1|1)-dimensional complex
supermanifolds. In the local approach their gluing transition functions are
superconformal transformations CRANE AND RABIN [1988], FRIEDAN [1986].
They appear as a result of the special reduction of the structure supergroup
GIDDINGS AND NELSON [1988]. We propose an alternative tangent space
reduction, which leads to new transformations twisting parity of the tangent space
DUPLIJ [1991a, 1996b].
We use the functional approach to superspace ROGERS [1980] which admits
existence of nontrivial topology in odd directions RABIN AND CRANE [1985b] and
can be suitable for physical applications BRUZZO [1986], VANDYK [1990].
20

Locally (1|1)-dimensional superspace C1|1
is Z = (z, θ), where z is an even
coordinate and θ is an odd one. The superanalytic (SA) transformation
TSA : C1|1
→ C1|1
is
˜z = f (z) + θ ∙ χ (z) ,
˜θ = ψ (z) + θ ∙ g (z) ,
(2.1)
where f (z) , g (z) : C1|0
→ C1|0
and ψ (z) , χ (z) : C1|0
→ C0|1
satisfy
supersmooth conditions generalizing C∞
ROGERS [1980]. The set of
transformations (2.1) form a semigroup of superanalytic transformations TSA
DUPLIJ [1991b]. The invertible ones form a subgroup, while the noninvertible
ones form an ideal DUPLIJ [1997b].
The invertibility of the superanalytic transformation (2.1) is determined first of all
by invertibility of the even functions f (z) and g (z)( because odd functions are
noninvertible by definition).
21

The tangent superspace in C1|1
is {∂, D}, where D = ∂θ + θ∂, ∂θ =
∂/∂θ, ∂ = ∂/∂z. The dual cotangent space is spanned by 1-forms {dZ, dθ},
where dZ = dz + θdθ (the signs as in CRANE AND RABIN [1988]). The
supersymmetry D2
= ∂, dZ2
= dz.
The semigroup of SA transformations TSA acts in the tangent and cotangent
superspaces by means of the tangent space matrix PA as

∂
D

 = PA


˜∂
˜D

 and d ˜Z, d˜θ = dZ, dθ PA, where
PA =


∂˜z − ∂˜θ ∙ ˜θ ∂˜θ
D˜z − D˜θ ∙ ˜θ D˜θ

 . (2.2)
In case of invertible SA transformations the matrix PA defines structure of a
supermanifold GIDDINGS AND NELSON [1988].
22

Indeed, if D˜θ = 0 (where is the soul map) we get for the superanalog of
Jacobian, the Berezinian BEREZIN [1987]
Ber PA =
∂˜z − ∂˜θ ∙ ˜θ
D˜θ
+
D˜z − D˜θ ∙ ˜θ ∂˜θ
D˜θ
2 . (2.3)
Using the Berezinian addition theorem DUPLIJ [1996b] we have
Ber PA = Ber PS + Ber PT , (2.4)
where
PS
def
=


∂˜z − ∂˜θ ∙ ˜θ ∂˜θ
0 D˜θ

 , (2.5)
PT
def
=


0 ∂˜θ
D˜z − D˜θ ∙ ˜θ D˜θ

 . (2.6)
23

Let us apply the conditions
Q
def
= ∂˜z − ∂˜θ ∙ ˜θ = 0, (2.7)
Δ
def
= D˜z − D˜θ ∙ ˜θ = 0 (2.8)
to the matrices PS and PT , then we derive
PSCf
def
= PS|Δ=0 (2.9)
PT P t
def
= PT |Q=0. (2.10)
The condition Δ = 0 (2.8) gives us the superconformal (SCf ) transformations
TSCf CRANE AND RABIN [1988], and the reduced matrix PSCf (2.9) is the
standard reduction of structure supergroup GIDDINGS AND NELSON [1988].
24

We consider another condition Q = 0 (2.7), which leads to the degenerated
transformations TT P t twisting parity of the standard tangent space (TPt )
DUPLIJ [1991b].
The alternative reduction DUPLIJ [1996b] of the tangent space supermatrix PA
gives us the supermatrix PT P t (2.10). The dual role of SCf and TPt
transformations is seen from the Berezinian addition theorem (2.4) DUPLIJ
[1996b].
Since SCf transformations give us the standard (even) superanalog of complex
structure LEVIN [1987], SCHWARZ [1994], we treat TPt transformations as
alternative odd N = 1 superanalog of complex structure DUPLIJ [1997a].
25

Using the projections (2.9) and (2.10) we have for the Berezinian
Ber PA =



Ber PSCf , Δ = 0,
Ber PT P t, Q = 0.
(2.11)
A general relation between Q and Δ is Q − DΔ = D˜θ
2
. After
corresponding projections we have
Q|Δ=0 = D˜θ
2
, (SCf ), (2.12)
Δ|Q=0 ≡ Δ0 = ∂θ ˜z − ∂θ
˜θ ∙ ˜θ, (TPt ). (2.13)
26

Using (2.12) one obtains GIDDINGS AND NELSON [1988]
PSCf =


D˜θ
2
∂˜θ
0 D˜θ

 . (2.14)
If ε D˜θ = 0 the Berezinian has the standard form
Ber PSCf = D˜θ. (2.15)
Remark 2.1. In case ε D˜θ = 0 the Berezinian cannot be defined and all, but
we can accept (2.15) as a definition of the Jacobian of noninvertible
superconformal transformations DUPLIJ [1990, 1996a].
27

For TPt (twisting parity of tangent space) transformations, using (2.13), we derive
PT P t =


0 ∂˜θ
∂θ ˜z − ∂θ
˜θ ∙ ˜θ D˜θ

 (2.16)
If ε D˜θ = 0 the Berezinian of PT P t can be determined as
Ber PT P t =
Δ0 ∙ ∂˜θ
D˜θ
2 =
∂Δ0 ∙ Δ0
2 D˜θ
3 . (2.17)
Since Δ0 is odd and so nilpotent, Ber PT P t is nilpotent, also
Ber PT P t = D
D˜z
D˜θ
(2.18)
which should be remarkably compared with (2.15) Ber PSCf = D˜θ.
28

Show manifestly the intriguing peculiarity of TPt transformations: twisting the
parity of tangent and cotangent spaces in the standard basis
SCf:



D = D˜θ ∙ ˜D,
d ˜Z = D˜θ
2
∙ dZ,
TPt:



∂ = ∂˜θ ∙ ˜D,
d ˜Z = Δ0 ∙ dθ.
(2.19)
The reduction conditions (2.7) and (2.8) fix 2 of 4 component functions form (2.1)
in each case. Usually CRANE AND RABIN [1988] SCf transformations TSCf are
parametrized by (f, ψ), while other functions are found from (2.7) and (2.8).
However, the latter can be done for invertible transformations only. To avoid this
difficulty we introduce an alternative parametrization by the pair (g, ψ), which
allows us to consider SCf and TPt transformations in a unified way and include
noninvertibility.
29

Indeed, fixing g (z) and ψ (z) we find



fn (z) = ψ (z) ψ (z) + 1+n
2 g2
(z) ,
χn (z) = g (z) ψ (z) + ng (z) ψ (z) ,
(2.20)
where n =



+1, SCf,
−1, TPt,
can be treated as a projection of some “reduction
spin” switching the type of transformation. The unified multiplication law is


h
ϕ


n
∗


g
ψ


m
=


g ∙ h ◦ fm + χm ∙ ψ ∙ h ◦ fm + χm ∙ ϕ ◦ fm
ϕ ◦ fm + ψ ∙ h ◦ fm

 ,
(2.21)
where (∗) is transformation composition and (◦) is the function composition.
30

For “reduction spin” projections we have only two definite products
(+1) ∗ (+1) = (+1) and (+1) ∗ (−1) = (−1). The first formula is a
consequence of PS ∙ PS ⊆ PS (see (2.5)), PS ∙ PS ⊆ PS it also follows the
standard cocycle condition CRANE AND RABIN [1988] ˜TSCf ∗ TSCf = ˜TSCf
on triple overlaps U ∩ ˜U ∩ ˜U, where U, ˜U, ˜U are open superdomains and
T : U → ˜U, ˜T : ˜U → ˜U, ˜T : U → ˜U.
In the invertible SCf case the cocycle condition leads to the definition of a super
Riemann surface as a holomorphic (1|1)-dimensional supermanifold equipped
with an additional one-dimensional subbundle CRANE AND RABIN [1988],
GIDDINGS AND NELSON [1988], LEVIN [1987], which grounds on the cocycle
relation D˜θ = D˜θ ∙ ˜D˜θ . However, TPt transformations TT P t form a
subsemigroup only providing additional conditions on component functions
DUPLIJ [1991b].
31

They have also another important abstract meaning: using the unrestricted
relation PT ∙ PS ⊆ PT we obtain a ”mixed cocycle condition”
˜TSCf ∗ TT P t = ˜TT P t, which gives the ”mixed cocycle relation”
∂˜θ = ∂˜θ ∙ ˜D˜θ. (2.22)
By analogy with SCf transformations, they can be used for constructing new
objects analogous to super Riemann surfaces and could possibly give additional
contributions to the fermionic string amplitude.
32

3 Ternary supersymmetry
3 Ternary supersymmetry
The Z3-graded analogue of Grassman algebra was introduced in ABRAMOV
ET AL. [1997]. Consider the associative algebra A3 over C spanned by N
generators θA
which satisfy the cubic relations
θA
θB
θC
= j θB
θC
θA
= j2
θC
θA
θB
, (3.1)
with j = e2iπ/3
, the primitive root of unity j3
= 1. The N2
products θA
θB
are
linearly independent entities, (θA
)3
= 0 . The algebra admits a natural
Z3-grading: the grades add up modulo 3; the numbers are grade 0, the
generators θA
are grade 1; the binary products are grade 2, and the ternary
products grade 0. The dimensions of the subsets of grade 0, 1 and 2 are,
respectively, N for grade 1, N2
for grade 2 and (N3
− N)/3 + 1 for grade 0.
Therefore the dimension of algebra A3 is N(N + 1)(N + 2)/3 + 1.
33

4 Polyadic analogs of integer number ring Z and field Z/pZ
The theory of finite fields LIDL AND NIEDERREITER [1997] plays a very important
role. Peculiarities of finite fields: 1) Uniqueness - they can have only special
numbers of elements (the order is any power of a prime integer pr
) and this fully
determines them, all finite fields of the same order are isomorphic; 2) Existence of
their “minimal” (prime) finite subfield of order p, which is isomorphic to the
congruence class of integers Z pZ.
We propose a special - version of the (prime) finite fields: instead of the binary
ring of integers Z, we consider a polyadic ring. The concept of the polyadic
integer numbers Z(m,n) as representatives of a fixed congruence class, forming
the (m, n)-ring (with m-ary addition and n-ary multiplication), was introduced in
DUPLIJ [2017a]. We define new secondary congruence classes and the
corresponding finite (m, n)-rings Z(m,n) (q) of polyadic integer numbers, which
give Z qZ in the “binary limit”. We construct the prime polyadic fields
F(m,n) (q), which can be treated as polyadic analog of the Galois field GF (p).
34

Notations
A polyadic (m, n)-ring is Rm,n = R | νm, μn , where R is a set, equipped
with m-ary addition νm : Rm
→ R and n-ary multiplication μn : Rn
→ R
which are connected by the polyadic distributive law: R | νm is a commutative
m-ary group and R | μn is a semigroup. A polyadic ring is derived, if νm and
μn are equivalent to a repetition of the binary addition and multiplication, while
R | + and R | ∙ are commutative (binary) group and semigroup.
An n-admissible “length of word (x)” should be congruent to 1 mod (n − 1),
containing μ (n − 1) + 1 elements ( μ is a “number of multiplications”)
μ
( μ)
n [x] (x ∈ R μ(n−1)+1
), or polyads. An m-admissible “quantity of words
(y)” in a polyadic “sum” has to be congruent to 1 mod (m − 1), i.e. consisting
of ν (m − 1) + 1 summands ( ν is a “number of additions”) ν
( ν )
m [y]
(y ∈ R ν (m−1)+1
).
“Polyadization” of a binary expression (m = n = 2): the multipliers
μ + 1 → μ (n − 1) + 1 and summands ν + 1 → ν (m − 1) + 1.
35

Example 4.1. “Trivial polyadization”: the simplest (m, n)-ring derived from the
ring of integers Z as the set of ν (m − 1) + 1 “sums” of n-admissible
( μ (n − 1) + 1)-ads (x), x ∈ Z μ(n−1)+1
LEESON AND BUTSON [1980].
The additive m-ary polyadic power and multiplicative n-ary polyadic power are
(inside polyadic products we denote repeated entries by
k
x, . . . , x as xk
)
x ν +m = ν( ν )
m x ν (m−1)+1
, x μ ×n = μ( μ)
n x μ(n−1)+1
, x ∈ R,
(4.1)
Polyadic and ordinary powers differ by 1: x ν +2 = x ν +1
, x μ ×2 = x μ+1
.
The polyadic idempotents in Rm,n satisfy
x ν +m = x, x μ ×n = x, (4.2)
and are called the additive ν-idempotent and the multiplicative μ-idempotent,
respectively. The idempotent zero z ∈ R, is (if it exists) defined by
νm [x, z] = z, ∀x ∈ Rm−1
. If a zero exists, it is unique. And x is nilpotent, if
x 1 +m = z, and all higher powers of a nilpotent element are nilpotent.
36

The unit e of Rm,n is multiplicative 1-idempotent μn en−1
, x = x, ∀x ∈ R
(in case of a noncommutative polyadic ring) x can be on any place.
In distinction with the binary case there are unusual polyadic rings :
1) with no unit and no zero (zeroless, nonunital);
2) with several units and no zero;
3) with all elements are units.
In polyadic rings invertibility is not connected with unit and zero elements.
For a fixed element x ∈ R its additive querelement ˜x and multiplicative
querelement ˉx are defined by
νm xm−1
, ˜x = x, μn xn−1
, ˉx = x, (4.3)
Because R | νm is a commutative group, each x ∈ R has its additive
querelement ˜x (and is querable or “polyadically invertible”). The n-ary semigroup
R | μn can have no multiplicatively querable elements at all. However, if every
x ∈ R has its unique querelement, then R | μn is an n-ary group.
37

Denote R∗
= R {z}, if the zero z exists. If R∗
| μn is the n-ary group, then
Rm,n is a (m, n)-division ring.
Definition 4.2. A commutative (m, n)-division ring is a (m, n)-field Fm,n.
The nonderived (m, n)-fields are in
Example 4.3. a) The set iR with i2
= −1 is a (2, 3)-field with a zero and no
unit (operations in C), but the multiplicative querelement of ix is −i x (x = 0).
b) The set of fractions ix/y | x, y ∈ Zodd
, i2
= −1 is a (3, 3)-field with no
zero and no unit (operations are in C), while the additive and multiplicative
querelements of ix/y are −ix/y and −iy/x, respectively.
c) The set of antidiagonal 2 × 2 matrices over R is a (2, 3)-field with zero
z =


0 0
0 0

 and two units e = ±


0 1
1 0

, but the unique
querelement of


0 x
y 0

 is


0 1/y
1/x 0

.
38

Ring of polyadic integer numbers Z(m,n)
The ring of polyadic integer numbers Z(m,n) was introduced in DUPLIJ [2017a].
Consider a congruence class (residue class) of an integer a modulo b
[[a]]b = {{a + bk} | k ∈ Z, a ∈ Z+, b ∈ N, 0 ≤ a ≤ b − 1} . (4.4)
Denote a representative by xk = x
[a,b]
k = a + bk, where {xk} is an infinite set.
Informally, there are two ways to equip (4.4) with operations:
1. The “External” way: to define operations between the classes [[a]]b. Denote
the class representative by [[a]]b ≡ a , and introduce the binary operations
+ , ∙ as
a1 + a2 = (a1 + a2) , (4.5)
a1 ∙ a2 = (a1a2) . (4.6)
39

The binary residue class ring is defined by
Z bZ = {{a } | + , ∙ , 0 , 1 } . (4.7)
With prime b = p, the ring Z pZ is a binary finite field having p elements.
2. The “Internal” way is to introduce (polyadic) operations inside a class [[a]]b
(with both a and b fixed). We define the commutative m-ary addition and
commutative n-ary multiplication of representatives xki
in the fixed
congruence class [[a]]b by
νm [xk1
, xk2
, . . . , xkm
] = xk1
+ xk2
+ . . . + xkm
, (4.8)
μn [xk1 , xk2 , . . . , xkn ] = xk1 xk2 . . . xkn , xki ∈ [[a]]b , ki ∈ Z.
(4.9)
Remark 4.4. Binary sums xk1
+ xk2
and products xk1
xk2
are not in [[a]]b.
40

Proposition 4.5 ( DUPLIJ [2017b]). The polyadic operations νm and μn
closed in [[a]]b, if the arities (m, n) have the minimal values satisfying
ma ≡ a (mod b) , (4.10)
an
≡ a (mod b) . (4.11)
Remark 4.6. If n = b = p is prime, then (4.11) is valid for any a ∈ N, which
is another formulation of Fermat’s little theorem.
Polyadic distributivity is inherited from that of Z, and therefore we have
Definition 4.7 ( DUPLIJ [2017a]). The congruence class [[a]]b equipped with
a structure of nonderived infinite commutative polyadic ring is called a
(m, n)-ring of polyadic integer numbers
Z(m,n) ≡ Z
[a,b]
(m,n) = {[[a]]b | νm, μn} . (4.12)
Example 4.8. In the residue class
[[3]]4 = {. . . − 25, −21, −17, −13, −9, −5, −1, 3, 7, 11, 15, 19, 23, 27, 31 . . .}
(4.13)
41

To retain the same class, we can add 4 ν + 1 = 5, 9, 13, 17, . . .
representatives and multiply 2 μ + 1 = 3, 5, 7, 9, . . . representatives only.
E.g., take “number” of additions ν = 2, and multiplications μ = 3, to get
(7 + 11 + 15 + 19 + 23) − 5 − 9 − 13 − 17 = 31 = 3 + 4 ∙ 7 ∈ [[3]]4,
((7 ∙ 3 ∙ 11) ∙ 19 ∙ 15) ∙ 31 ∙ 27 = 55 103 895 = 3 + 4 ∙ 13775973 ∈ [[3]]4.
This means that [[3]]4 is the polyadic (5, 3)-ring Z(5,3) = Z
[3,4]
(5,3).
Remark 4.9. Elements of the (m, n)-ring Z(m,n)(polyadic integer numbers) are
not ordinary integers (forming a (2, 2)-ring). A representative x
[a,b]
k , e.g.
3 = 3(5,3) ∈ Z
[3,4]
(5,3) is different from 3 = 3(3,2) ∈ Z
[1,2]
(3,2), and different from the
binary 3 ∈ Z ≡ Z
[0,1]
(2,2), i.e. properly speaking 3(5,3) = 3(3,2) = 3, since their
operations (multiplication and addition) are different.
Definition 4.10. A polyadic prime number is such that obeys a unique expansion
xkp
= μ( )
n xkp
, e (n−1)
, (4.14)
where e a polyadic unit of Z(m,n) (if exists).
42

The parameters-to-arity mapping
Remark 4.11. a) Solutions to (4.10) and (4.11) 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 4.12. The parameters-to-arity mapping ψ : (a, b) −→ (m, n) is a
partial surjection.
The characterization of the fixed congruence class [[a]]b and the corresponding
(m, n)-ring of polyadic integer numbers Z
[a,b]
(m,n) can be done in terms of the
shape invariants I, J ∈ Z+ defined uniquely by (TABLE 3 in DUPLIJ [2017a])
I = I[a,b]
m = (m − 1)
a
b
, J = J[a,b]
n =
an
− a
b
. (4.15)
In the binary case, when m = n = 2 (a = 0, b = 1), both shape invariants
vanish, I = J = 0. There exist “partially” binary cases, when only n = 2 and
m = 2, while J is nonzero, for instance in Z
[6,10]
(6,2) we have I = J = 3.
43

Polyadic rings of secondary classes
A special method of constructing a finite nonderived polyadic ring by combining
the “External” and “Internal” methods was given in DUPLIJ [2017b].
Introduce the finite polyadic ring Z(m,n) cZ, where Z(m,n) is a polzadic ring.
If we directly consider the “double” class {a + bk + cl} and fix a and b, then the
factorization by cZ will not give a closed operations for arbitrary c.
Assertion 4.13. If the finite polyadic ring Z
[a,b]
(m,n) cZ has q elements, then
c = bq. (4.16)
Definition 4.14. A secondary (equivalence) class of a polyadic integer
x
[a,b]
k = a + bk ∈ Z
[a,b]
(m,n) “modulo” bq (with q being the number of
representatives x
[a,b]
k , for fixed b ∈ N and 0 ≤ a ≤ b − 1) is
x
[a,b]
k
bq
= {{(a + bk) + bql} | l ∈ Z, q ∈ N, 0 ≤ k ≤ q − 1} .
(4.17)
44

Remark 4.15. In the binary limit a = 0, b = 1 and Z
[0,1]
(2,2) = Z, the secondary
class becomes the ordinary class (4.4).
If the values of a, b, q are clear from the context, we denote the secondary class
representatives by an integer with two primes x
[a,b]
k
bq
≡ xk ≡ x .
Example 4.16. a) For a = 3, b = 6 and for 4 elements and k = 0, 1, 2, 3
x
[3,6]
k
24
= 3 , 9 , 15 , 21 , ([[k]]4 = 0 , 1 , 2 , 3 ) . (4.18)
b) If a = 4, b = 5, for 3 elements and k = 0, 1, 2 we get
x
[4,5]
k
15
= 4 , 9 , 14 , ([[k]]3 = 0 , 1 , 2 ) . (4.19)
45

c) Let a = 3, b = 5, then for q = 4 elements we have the secondary classes
with k = 0, 1, 2, 3 (the binary limits are in brackets)
x
[3,5]
k
20
= 3 , 8 , 13 , 18 =



3 = {. . . − 17, 3, 23, 43, 63, . . .} ,
8 = {. . . − 12, 8, 28, 48, 68, . . .} ,
13 = {. . . − 7, 13, 33, 53, 73, . . .} ,
18 = {. . . − 2, 18, 38, 58, 78, . . .} ,
(4.20)







[[k]]4 = 0 , 1 , 2 , 3 =



0 = {. . . − 4, 0, 4, 8, 12, . . .} ,
1 = {. . . − 3, 1, 5, 9, 13, . . .} ,
2 = {. . . − 2, 2, 6, 10, 14, . . .} ,
3 = {. . . − 1, 3, 7, 11, 15, . . .} .







(4.21)
Difference: 1) they are described by rings of different arities; 2) some of them are
fields.
46

Finite polyadic rings
Now we determine the nonderived polyadic operations between secondary
classes which lead to finite polzadic rings.
Proposition 4.17. The set {xk} of q secondary classes k = 0, . . . , q − 1 (with
the fixed a, b) can be endowed with the following commutative m-ary addition
xkadd
= νm xk1
, xk1
, . . . , xkm
, (4.22)
kadd ≡ (k1 + k2 + . . . + km) + I[a,b]
m (mod q) (4.23)
and commutative n-ary multiplication
xkmult
= μn xk1
, xk1
, . . . , xkn
, (4.24)
kmult ≡ an−1
(k1 + k2 + . . . + kn) + an−2
b (k1k2 + k2k3 + . . . + kn−1kn) + . . .
+bn−1
k1 . . . kn + J[a,b]
n (mod q) , (4.25)
satisfying the polyadic distributivity, shape invariants I
[a,b]
m , J
[a,b]
n are in (4.15).
47

Definition 4.18. The set of secondary classes (4.17) equipped with operations
(4.22), (4.24) is denoted by
Z(m,n) (q) ≡ Z
[a,b]
(m,n) (q) = Z
[a,b]
(m,n) (bq) Z = {{xk} | νm, μn} , (4.26)
and is a finite secondary class (m, n)-ring of polyadic integer numbers
Z(m,n) ≡ Z
[a,b]
(m,n). The value q (the number of elements) is called its order.
Example 4.19. a) In (5, 3)-ring Z
[3,4]
(4,3) (2) with 2 secondary classes all elements
are units (marked by subscript e) e1 = 3e = 3 , e2 = 7e = 7 , because
μ3 [3 , 3 , 3 ] = 3 , μ3 [3 , 3 , 7 ] = 7 , μ3 [3 , 7 , 7 ] = 3 , μ3 [7 , 7 , 7 ] = 7 .
(4.27)
b) The ring Z
[5,6]
(7,3) (4) consists of 4 units
e1 = 5e , e2 = 11e , e3 = 17e , e4 = 23e , and no zero.
Remark 4.20. Equal arity finite polyadic rings of the same order Z
[a1,b1]
(m,n) (q) and
Z
[a2,b2]
(m,n) (q) may be not isomorphic.
48

Example 4.21. The finite polyadic ring Z
[1,3]
(4,2) (2) of order 2 consists of unit
e = 1e = 1 and zero z = 4z = 4 only,
μ2 [1 , 1 ] = 1 , μ2 [1 , 4 ] = 4 , μ2 [4 , 4 ] = 4 , (4.28)
and therefore Z
[1,3]
(4,2) (2) is a field, because {1 , 4z } 4z is a (trivial) binary
group, consisting of one element 1e .
However, Z
[4,6]
(4,2) (2) has the zero z = 4z = 4 , 10 and has no unit, because
μ2 [4 , 4 ] = 4 , μ2 [4 , 10 ] = 4 , μ2 [10 , 10 ] = 4 , (4.29)
so that Z
[4,6]
(4,2) (2) is not a field, because the nonzero element 10 is nilpotent.
Their additive 4-ary groups are also not isomorphic, while Z
[1,3]
(4,2) (2) and
Z
[4,6]
(4,2) (2) have the same arity (m, n) = (4, 2) and order 2.
Assertion 4.22. 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.
49

Finite polyadic fields
Proposition 4.23. A finite polyadic ring Z
[a,b]
(m,n) (q) is a secondary class finite
(m, n)-field F
[a,b]
(m,n) (q) if all its elements except z (if it exists) are polyadically
multiplicative invertible having a unique querelement.
In the binary case LIDL AND NIEDERREITER [1997] the residue (congruence)
class ring (4.7) with q elements Z qZ is a congruence class (non-extended)
field, if its order q = p is a prime number, such that
F (p) = [[a]]p | + , ∙ , 0 , 1 , a = 0, 1, . . . , p − 1.
All non-extended binary fields of a fixed prime order p are isomorphic, and so it is
natural to study them in a more “abstract” way. The mapping Φp [[a]]p = a is
an isomorphism of binary fields Φp : F (p) → F (p), where
F (p) = {{a} | +, ∙, 0, 1}mod p is an “abstract” non-extended (prime) finite field
of order p (or Galois field GF (p)).
50

Consider the set of polyadic integer numbers
{xk} ≡ x
[a,b]
k = {a + bk} ∈ Z
[a,b]
(m,n), b ∈ N and
0 ≤ a ≤ b − 1, 0 ≤ k ≤ q − 1, q ∈ N, which obey the operations (4.8)–(4.9).
Definition 4.24. The “abstract” non-extended finite (m, n)-field of order q is
F(m,n) (q) ≡ F
[a,b]
(m,n) (q) = {{a + bk} | νm, μn}mod bq , (4.30)
if {{xk} | νm}mod bq is an additive m-ary group, and {{xk} | μn}mod bq (or,
when zero z exists, {{xk z} | μn}mod bq) is a multiplicative n-ary group.
Define a one-to-one onto mapping from the secondary congruence class to its
representative by Φ
[a,b]
q x
[a,b]
k
bq
= x
[a,b]
k and arrive
Proposition 4.25. The mapping Φ
[a,b]
q : F
[a,b]
(m,n) (q) → F
[a,b]
(m,n) (q) is a
polyadic ring homomorphism (being, in fact, an isomorphism).
In TABLE 1 we present the “abstract” non-extended polyadic finite fields
F
[a,b]
(m,n) (q) of lowest arity shape (m, n) and orders q.
51

Table 1: The finite polyadic rings Z
[a,b]
(m,n) (q) and (m, n)-fields F
[a,b]
m,n (q).
a b 2 3 4 5 6
1
m = 3
n = 2
1e,3
1e,3z,5
1e,3,5,7
q=5,7,8
m = 4
n = 2
1e,4z
1e,4,7
1e,4z,7,10
q=5,7,9
m = 5
n = 2
1e,5
1e,5,9z
1e,5,9,13
q=5,7,8
m = 6
n = 2
1e,6z
1e,6z,11
1e,6,11,16z
q=5,7
m = 7
n = 2
1e,7
1e,7,13
1e,7,13,19
q=5,6,7,8,9
2
m = 4
n = 3
2z,5e
2,5,8e
2,5e,8z,11e
q=5,7,9
m = 6
n = 5
2z,7e
2e,7,12z
2,7e,12z,17e
q=5,7
m = 4
n = 3
2,8z
2,8e,14
2,8z,14,20
q=5,7,9
3
m = 5
n = 3
3e,7e
3z,7e,11e
3,7e,11,15e
q=5,6,7,8
m = 6
n = 5
3e,8z
3z,8e,13e
3e,8z,13e,18
q=5,7
m = 3
n = 2
3,9e
3,9z,15
3,9e,15,21
q=5,7,8
52

In the multiplicative structure the following crucial differences between the binary
finite fields F (q) and polyadic fields F(m,n) (q) can be outlined.
1. The order of a non-extended finite polyadic field may not be prime (e.g.,
F
[1,2]
(3,2) (4), F
[3,4]
(5,3) (8), F
[2,6]
(4,3) (9)), and may not even be a power of a prime
binary number (e.g. F
[5,6]
(7,3) (6), F
[3,10]
(11,5) (10)), and see TABLE 2.
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 2 by subscripts
which indicate the number of units.
4. The(m, n)-fields can be zeroless-nonunital, but have unique additive and
multiplicative querelements: [[a]]b | νm , [[a]]b | μn are polyadic groups.
5. The zeroless-nonunital polyadic fields are totally (additively and
multiplicatively) nonderived.
53

Example 4.26. 1) The zeroless-nonunital polyadic finite fields having lowest
|a + b| are, e.g., F
[3,8]
(9,3) (2), F
[3,8]
(9,3) (4), F
[5,8]
(9,3) (4), F
[5,8]
(9,3) (8), also F
[4,9]
(10,4) (3),
F
[4,9]
(10,4) (9), and F
[7,9]
(10,4) (3), F
[7,9]
(10,4) (9).
2) The multiplication of the zeroless-nonunital (9, 3)-field F
[5,8]
(9,3) (2) is
μ3 [5, 5, 5] = 13, μ3 [5, 5, 13] = 5, μ3 [5, 13, 13] = 13, μ3 [13, 13, 13] = 5.
The (unique) multiplicative querelements ˉ5 = 13, 13 = 5. The addition is
ν9 5
9
= 13, ν9 5
8
, 13 = 5, ν9 5
7
, 13
2
= 13, ν9 5
6
, 13
3
= 5, ν9 5
5
, 13
4
= 13,
ν9 5
4
, 13
5
= 5, ν9 5
3
, 13
6
= 13, ν9 5
2
, 13
7
= 5, ν9 5, 13
8
= 13, ν9 13
9
= 5.
The additive (unique) querelements are ˜5 = 13, 13 = 5. So all elements are
additively and multiplicatively querable (polyadically invertible), and therefore ν9
is 9-ary additive group operation and μ3 is 3-ary multiplicative group operation,
as it should be for a field. Because it contains no unit and no zero, F
[5,8]
(9,3) (2) is
actually a zeroless-nonunital finite (9, 3)-field of order 2.
54

Example 4.27. The (4, 3)-ring Z
[2,3]
(4,3) (6) is zeroless, and [[3]]4 | ν4 is its
4-ary additive group (each element has a unique additive querelement). Despite
each element of [[2]]3 | μ3 having a querelement, it is not a multiplicative 3-ary
group, because for the two elements 2 and 14 we have nonunique querelements
μ3 [2, 2, 5] = 2, μ3 [2, 2, 14] = 2, μ3 [14, 14, 2] = 14, μ3 [14, 14, 11] = 14.
(4.31)
Example 4.28. 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 2).
Despite both being zeroless, the first F
[5,8]
(9,3) (q) are nonunital, while the second
F
[7,8]
(9,3) (q) has two units, which makes an isomorphism impossible.
55

Polyadic field order
In binary case the order of an element x ∈ F (p) is defined as a smallest integer
λ such that xλ
= 1. Obviously, the set of fixed order elements forms a cyclic
subgroup of the multiplicative binary group of F (p), and λ | (p − 1). If
λ = p − 1, such an element is called a primitive (root), it generates all elements,
and these exist in any finite binary field. Any element of F (p) is idempotent
xp
= x, while all its nonzero elements satisfy xp−1
= 1 (Fermat’s little
theorem). A non-extended (prime) finite field is fully determined by its order p (up
to isomorphism), and, moreover, any F (p) is isomorphic to Z pZ.
In the polyadic case, the situation is more complicated. Because the related
secondary class structure (4.30) contains parameters in addition to the number of
elements q, the order of (non-extended) polyadic fields may not be prime, or nor
even a power of a prime integer (e.g. F
[5,6]
(7,3) (6) or F
[3,10]
(11,5) (10)). Because finite
polyadic fields can be zeroless, nonunital and have many (or even all) units (see
TABLE 2), we cannot use units in the definition of the element order.
56

Definition 4.29. If an element of the finite polyadic field x ∈ F(m,n) (q) satisfies
x λp ×n = x, (4.32)
then the smallest such λp is called the idempotence polyadic order ord x = λp.
Definition 4.30. The idempotence polyadic order λ[a,b]
p of a finite polyadic field
F
[a,b]
(m,n) (q) is the maximum λp of all its elements, we call such field
λ[a,b]
p -idempotent and denote ord F
[a,b]
(m,n) (q) = λ[a,b]
p .
In TABLE 2 we present the idempotence polyadic order λ[a,b]
p for small a, b.
Definition 4.31. Denote by q∗ the number of nonzero distinct elements in
F(m,n) (q)
q∗ =



q − 1, if ∃z ∈ F(m,n) (q)
q, if z ∈ F(m,n) (q) ,
(4.33)
which is called a reduced (field) order (in binary case we have the first line only).
57

Table 2: Idempotence polyadic orders λ[a,b]
p for finite polyadic fields F
[a,b]
(m,n) (q).
[[a]]b Arities Finite polyadic field order q
b a (m, n) 2 3 4 5 6 7 8 9 10
2 1 (3, 2) 2 2 2 4 ∅ 6 4 ∅ ∅
3 1 (4, 2) 1 3 ∅ 4 ∅ 6 ∅ 9 ∅
2 (4, 3) 1 3 ∅ 22e ∅ 32e ∅ 9 ∅
4 1 (5, 2) 2 2 4 4 ∅ 6 8 ∅ ∅
3 (5, 3) 12e 12e 22e 22e ∅ 32e 42e ∅ ∅
5 1 (6, 2) 1 2 ∅ 5 ∅ 6 ∅ ∅ ∅
2 (6, 5) 1 12e ∅ 5 ∅ 32e ∅ ∅ ∅
3 (6, 5) 1 12e ∅ 5 ∅ 32e ∅ ∅ ∅
4 (6, 3) 1 12e ∅ 5 ∅ 32e ∅ ∅ ∅
6 1 (7, 2) 2 3 2 4 6 6 4 9 ∅
2 (4, 3) ∅ 3 ∅ 22e ∅ 32e ∅ 9 ∅
3 (3, 2) 2 ∅ 2 4 ∅ 6 4 ∅ ∅
4 (4, 2) ∅ 3 ∅ 4 ∅ 6 ∅ 9 ∅
5 (7, 3) 12e 3 14e 22e 32e 32e 2 9 ∅
7 1 (8, 2) 1 2 ∅ 4 ∅ 7 ∅ ∅ ∅
2 (8, 4) 1 2 ∅ 4 ∅ 7 ∅ ∅ ∅
3 (8, 7) 1 12e ∅ 22e ∅ 7 ∅ ∅ ∅
4 (8, 4) 1 2 ∅ 4 ∅ 7 ∅ ∅ ∅
5 (8, 7) 1 12e ∅ 22e ∅ 7 ∅ ∅ ∅
6 (8, 3) 1 12e ∅ 22e ∅ 7 ∅ ∅ ∅
8 1 (9, 2) 2 2 4 4 ∅ 6 8 ∅ ∅
3 (9, 3) 2 12e 4 22e ∅ 32e 8 ∅ ∅58

Theorem 4.32. If a finite polyadic field F(m,n) (q) has an order q, such that
q∗ = qadm
∗ = (n − 1) + 1 is n-admissible, then (for n ≥ 3 and one unit):
1. A sequence of the length q∗ (n − 1) built from any fixed element
y ∈ F(m,n) (q) is neutral
μ(q∗)
n x, yq∗(n−1)
= x, ∀x ∈ F(m,n) (q) . (4.34)
2. Any element y satisfies the polyadic idempotency condition
y q∗ ×n = y, ∀y ∈ F(m,n) (q) . (4.35)
Finite polyadic fields F
[a,b]
(m,n) (q) having n-admissible reduced order
q∗ = qadm
∗ = (n − 1) + 1 ( ∈ N) (underlined in TABLE 2) are closest to the
binary finite fields F (p) in their general properties: they are always half-derived,
while if they additionally contain a zero, they are fully derived.
If q∗ = qadm
∗ , then F
[a,b]
(m,n) (q) can be nonunital or contain more than one unit
(subscripts in TABLE 2).
59

Assertion 4.33. The finite fields F
[a,b]
(m,n) (q) of n-admissible reduced order
q∗ = qadm
∗ cannot have more than one unit and cannot be zeroless-nonunital.
Assertion 4.34. If q∗ = qadm
∗ , and F
[a,b]
(m,n) (q) is unital zeroless, then the
reduced order q∗ is the product of the idempotence polyadic field order
λ[a,b]
p = ord F
[a,b]
(m,n) (q) and the number of units κe (if a b and n ≥ 3)
q∗ = λ[a,b]
p κe. (4.36)
Structure of the multiplicative group G
[a,b]
n (q∗) of F
[a,b]
(m,n) (q)
Some properties of commutative cyclic n-ary groups were considered for
particular relations between orders and arity. Here we have: 1) more parameters
and different relations between these, the arity m, n and order q; 2) the
(m, n)-field under consideration, which leads to additional restrictions. In such a
way exotic polyadic groups and fields arise which have unusual properties that
have not been studied before.
60

Definition 4.35. An element xprim ∈ G
[a,b]
n (q∗) is called n-ary primitive, if its
idempotence order is
λp = ord xprim = q∗. (4.37)
All λp polyadic powers x
1 ×n
prim , x
2 ×n
prim , . . . , x
q∗ ×n
prim ≡ xprim generate other
elements, and so G
[a,b]
n (q∗) is a finite cyclic n-ary group generated by xprim,
i.e. G
[a,b]
n (q∗) = x
i ×n
prim | μn . Number primitive elements in κprim.
Assertion 4.36. For zeroless F
[a,b]
(m,n) (q) and prime order q = p, we have
λ[a,b]
p = q, and G
[a,b]
n (q) is indecomposable (n ≥ 3).
Example 4.37. The smallest 3-admissible zeroless polyadic field is F
[2,3]
(4,3) (3)
with the unit e = 8e and two 3-ary primitive elements 2, 5 having 3-idempotence
order ord 2 = ord 5 = 3, so κprim = 2 , because
2 1 ×3 = 8e, 2 2 ×3 = 5, 2 3 ×3 = 2, 5 1 ×3 = 8e, 5 2 ×3 = 2, 5 3 ×3 = 5,
(4.38)
and therefore G
[2,3]
3 (3) is a cyclic indecomposable 3-ary group.
61

Assertion 4.38. If F
[a,b]
(m,n) (q) is zeroless-nonunital, then every element is n-ary
primitive, κprim = q, also λ[a,b]
p = q (the order q can be not prime), and
G
[a,b]
n (q) is a indecomposable commutative cyclic n-ary group without identity
(n ≥ 3).
Example 4.39. The (10, 7)-field F
[5,9]
(10,7) (9) is zeroless-nonunital, each element
(has λp = 9) is primitive and generates the whole field, and therefore
κprim = 9, thus the 7-ary multiplicative group G
[5,9]
7 (9) is indecomposable and
without identity.
The structure of G
[a,b]
n (q∗) can be extremely nontrivial and may have no analogs
in the binary case.
Assertion 4.40. If there exists more than one unit, then:
1. If G
[a,b]
n (q∗) can be decomposed on its n-ary subroups, the number of units
κe coincides with the number of its cyclic n-ary subgroups
G
[a,b]
n (q∗) = G1 ∪ G2 . . . ∪ Gke
which do not intersect Gi ∩ Gj = ∅,
i, j = i = 1, . . . , κe, i = j.
62

2. If a zero exists, then each Gi has its own unit ei, i = 1, . . . , κe.
3. In the zeroless case G
[a,b]
n (q) = G1 ∪ G2 . . . ∪ Gke ∪ E (G), where
E (G) = {ei} is the split-off subgroup of units.
Example 4.41. 1) In the (9, 3)-field F
[5,8]
(9,3) (7) there is a single zero z = 21z
and two units e1 = 13e, e2 = 29e, and so its multiplicative 3-ary group
G
[5,8]
3 (6) = {5, 13e, 29e, 37, 45, 53} consists of two nonintersecting (which is
not possible in the binary case) 3-ary cyclic subgroups G1 = {5, 13e, 45} and
G2 = {29e, 37, 53} (for both λp = 3)
G1 = 5 1 ×3 = 13e, 5 2 ×3 = 45, 5 3 ×3 = 5 , ˉ5 = 45, 45 = 5,
G2 = 37 1 ×3 = 29e, 37 2 ×3 = 53, 37 3 ×3 = 37 , 37 = 53, 53 = 37.
All nonunital elements in G
[5,8]
3 (6) are (polyadic) 1-reflections, because
5 1 ×3 = 45 1 ×3 = 13e and 37 1 ×3 = 53 1 ×3 = 29e, and so the subgroup
of units E (G) = {13e, 29e} is unsplit E (G) ∩ G1,2 = ∅.
63

2) For the zeroless F
[7,8]
(9,3) (8), its multiplicative 3-group
G
[5,8]
3 (6) = {7, 15, 23, 31e, 39, 47, 55, 63e} has two units e1 = 31e,
e2 = 63e, and it splits into two nonintersecting nonunital cyclic 3-subgroups
(λp = 4 and λp = 2) and the subgroup of units
G1 = 7 1 ×3 = 23, 7 2 ×3 = 39, 7 3 ×3 = 55, 7 4 ×3 = 4 ,
ˉ7 = 55, 55 = 7, 23 = 39, 39 = 23,
G2 = 15 1 ×3 = 47, 15 2 ×3 = 15 , 15 = 47, 47 = 15,
E (G) = {31e, 63e} .
There are no μ-reflections, and so E (G) splits out E (G) ∩ G1,2 = ∅.
If all elements are units E (G) = G
[a,b]
n (q), the group is 1-idempotent λp = 1.
Assertion 4.42. If F
[a,b]
(m,n) (q) is zeroless-nonunital, then there no n-ary cyclic
subgroups in G
[a,b]
n (q).
64

The subfield structure of F
[a,b]
(m,n) (q) can coincide with the corresponding
subgroup structure of the multiplicative n-ary group G
[a,b]
n (q∗), only if its additive
m-ary group has the same subgroup structure. However, we have
Assertion 4.43. Additive m-ary groups of all polyadic fields F
[a,b]
(m,n) (q) have the
same structure: they are polyadically cyclic and have no proper m-ary subgroups.
Therefore, in additive m-ary groups each element generates all other elements,
i.e. it is a primitive root.
Theorem 4.44. The polyadic field F
[a,b]
(m,n) (q), being isomorphic to the
(m, n)-field of polyadic integer numbers Z
[a,b]
(m,n) (q), has no any proper subfield.
In this sense, F
[a,b]
(m,n) (q) can be named a prime polyadic field.
65

5 Conclusion
5 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 Z pZ LIDL AND NIEDERREITER [1997].
Conjecture 5.1. A finite (m, n)-field (with m > n) should contain a minimal
subfield which is isomorphic to one of the prime polyadic fields constructed
above, and therefore the introduced here finite polyadic fields F
[a,b]
(m,n) (q) can be
interpreted as polyadic analogs of the prime Galois field GF (p).
ACKNOWLEDGEMENTS
(Polyadic part) The author would like to express his sincere thankfulness to
Joachim Cuntz, Christopher Deninger, Mike Hewitt, Grigorij Kurinnoj, Daniel Lenz,
Jim Stasheff, Alexander Voronov, and Wend Werner for fruitful discussions.
66

APPENDIX. Multiplicative properties of exotic finite polyadic fields
APPENDIX. 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
F
[5,6]
(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.
67

μ3 x7
= x, ∀x ∈ F
[5,6]
(7,3) (6). 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]
μ3 [5, 5, 35] = 11, μ3 [5, 11, 11] = 29, μ3 [5, 11, 17] = 35, μ3 [5, 11, 23] = 5, μ3 [5, 11
μ3 [5, 11, 35] = 17, μ3 [5, 17, 17] = 5, μ3 [5, 17, 23] = 11, μ3 [5, 17, 29] = 17, μ3 [5, 17
μ3 [5, 23, 23] = 17, μ3 [5, 23, 29] = 23, μ3 [5, 23, 35] = 29, μ3 [5, 29, 29] = 29, μ3 [5, 2
μ3 [5, 35, 35] = 5, μ3 [11, 11, 11] = 35, μ3 [11, 11, 17] = 5, μ3 [11, 11, 23] = 11, μ3 [11
μ3 [11, 11, 35] = 23, μ3 [11, 11, 17] = 5, μ3 [11, 17, 17] = 11, μ3 [11, 17, 23] = 17, μ3 [1
μ3 [11, 17, 35] = 29, μ3 [11, 17, 23] = 17, μ3 [11, 17, 29] = 23, μ3 [11, 17, 35] = 29, μ3 [
μ3 [11, 23, 29] = 29, μ3 [11, 23, 23] = 23, μ3 [11, 23, 35] = 35, μ3 [11, 29, 29] = 35, μ3 [
μ3 [11, 35, 35] = 11, μ3 [17, 17, 17] = 17, μ3 [17, 17, 23] = 23, μ3 [17, 17, 29] = 29, μ3 [
μ3 [17, 23, 23] = 29, μ3 [17, 23, 29] = 35, μ3 [17, 29, 29] = 5, μ3 [17, 29, 35] = 11, μ3 [1
μ3 [23, 23, 23] = 35, μ3 [23, 23, 29] = 5, μ3 [23, 23, 35] = 11, μ3 [23, 29, 29] = 11, μ3 [2
μ3 [23, 35, 35] = 23, μ3 [29, 29, 29] = 17, μ3 [29, 29, 35] = 23, μ3 [29, 35, 35] = 29, μ3 [
68

The multiplicative querelements are ˉ5 = 29, 29 = 5, 11 = 23, 23 = 11.
Because
5 1 ×3 = 17e, 5 2 ×3 = 29, 5 3 ×3 = 5, 29 1 ×3 = 17e, 29 2 ×3 = 5, 29 3 ×3 = 29
(A.1)
11 1 ×3 = 35e, 11 2 ×3 = 23, 11 3 ×3 = 11, 23 1 ×3 = 35e, 23 2 ×3 = 11, 23 3 ×3
(A.2)
the multiplicative 3-ary group G
[5,6]
(7,3) (6) consists of two nonintersecting cyclic
3-ary subgroups
G
[5,6]
(7,3) (6) = G1 ∪ G2, G1 ∩ G2 = ∅, (A.3)
G1 = {5, 17e, 29} , (A.4)
G2 = {11, 23, 35e} , (A.5)
which is impossible for binary subgroups, as these always intersect in the identity
of a binary group.
69

2) The finite polyadic field F
[5,6]
(7,3) (4) = {{5, 11, 17, 23} | ν7, 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]
μ3 [5, 11, 17] = 23, μ3 [5, 11, 23] = 17, μ3 [5, 17, 17] = 5, μ3 [5, 17, 23] = 11, μ3 [5, 23
μ3 [11, 11, 11] = 11, μ3 [11, 11, 17] = 17, μ3 [11, 11, 23] = 23, μ3 [11, 17, 17] = 11, μ3 [
μ3 [11, 23, 23] = 11, μ3 [17, 17, 17] = 17, μ3 [17, 17, 23] = 23, μ3 [17, 23, 23] = 17, μ3 [
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
shape. Indeed, consider these two (9, 3)-fields of order 2, that are F
[3,8]
(9,3) (2)
and F
[7,8]
(9,3) (2). The first is zeroless-nonunital, while the second is zeroless with
two units, i.e. all elements are units. The multiplication of F
[3,8]
(9,3) (2) is
μ3 [3, 3, 3] = 11, μ3 [3, 3, 11] = 3, μ3 [3, 11, 11] = 11, μ3 [11, 11, 11] = 3,
70

having the multiplicative querelements ˉ3 = 11, 11 = 3. For F
[7,8]
(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.
They have different idempotence polyadic orders ord F
[3,8]
(9,3) (2) = 2 and
ord F
[7,8]
(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.
4) The smallest exotic finite polyadic field with more than one unit is
F
[2,3]
(4,3) (5) = {{2, 5, 8, 11, 14} | ν4, μ3} of order 5 with two units
{11, 14} ≡ {11e, 14e} and the zero 5 ≡ 5z. The additive querelements are
˜2 = 11e, ˜8 = 14e, 11e = 8, 14e = 2. (A.6)
The idempotence polyadic order is ord F
[2,3]
(4,3) (5) = 2, because
2 2 ×3 = 2, 8 2 ×3 = 8, (A.7)
71

and their multiplicative querelements are ˉ2 = 8, ˉ8 = 2. The multiplication is
given by the cyclic 3-ary group G
[2,3]
3 (4) = {{2, 8, 11, 14} | μ3} as:
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]
μ3 [8, 8, 8] = 2, μ3 [8, 8, 11] = 14, μ3 [8, 8, 14] = 11, μ3 [8, 11, 11] = 8, μ3 [8, 11, 14]
μ3 [8, 14, 14] = 8, μ3 [11, 11, 11] = 11, μ3 [11, 11, 14] = 14, μ3 [11, 14, 14] = 11, μ3 [14
Despite having two units, the cyclic 3-ary group G
[2,3]
3 (4) has no decomposition
into nonintersecting cyclic 3-ary subgroups, as in (A.3). One of the reasons is that
the polyadic field F
[5,6]
(7,3) (6) is zeroless, while F
[2,3]
(4,3) (5) has a zero.
72

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