ON ENDOMORPHISMS OF THE ADDITIVE MONOID OF SUB-NETS OF A TWO-LAYER NEURAL NETWORK Текст научной статьи по специальности «Математика»

Серия «Математика» 2022. Т. 39. С. 111—126

Онлайн-доступ к журналу: http://mathizv.isu.ru

ИЗВЕСТИЯ

Иркутского государственного университета

Research article

УДК 512.577+519.68:007.5

MSC 08A35, 08A62, 68Q06, 94C11

DOI https://doi.org/10.26516/1997-7670.2022.39.111

On Endomorphisms of the Additive Monoid of Subnets of a Two-layer Neural Network

Andrey V. LitavrinlK

1 Siberian Federal University, Krasnoyarsk, Russian Federation K [email protected]

Abstract. Previously, for each multilayer neural network of direct signal propagation (hereinafter, simply a neural network), finite commutative groupoids were introduced, which were called additive subnet groupoids. These groupoids are closely related to the subnets of the neural network over which they are built. A grupoid is a monoid if and only if it is built over a two-layer neural network. Earlier, endomorphisms and their properties were studied for these groupoids. Some endomorphisms were constructed, but an exhaustive element-by-element description was not received. It was shown that every finite monoid is isomorphic to some submonoid of the monoid of all endomorphisms of a suitable additive subnet groupoid for some suitable neural network.

In this paper, we study endomorphisms of additive groupoids of subnets of two-layer neural networks. The main result of the work is an element-wise description of the monoid of all endomorphisms of additive monoids of subnets built over a two-layer neural network. The item-by-item description is obtained by constructing a general form of endomorphism. The general view of an endomorphism is parameterized by the endomorphisms of suitable booleans with respect to the union operation. Therefore, endomorphisms of these Booleans were studied in this work. In particular, the semirings of endomorphisms of these Booleans with respect to the union were studied. In addition, to describe the general form of the endomorphism of the additive monoid of subnets, homomorphisms of one Boalean into another (with respect to union) were used.

Keywords: groupoid endomorphism, feedforward multilayer neural network, multilayer neural network subnet

Acknowledgements: This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-02-2022-876).

For citation: Litavrin A. V.On Endomorphisms of the Additive Monoid of Subnets of a Two-layer Neural Network. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 39, pp. 111-126. https://doi.org/10.26516/1997-7670.2022.39.111

Научная статья

Об эндоморфизмах аддитивного моноида подсетей двухслойной нейронной сети

А. В. Литаврин1и

1 Сибирский федеральный университет, Красноярск, Российская Федерация И [email protected]

Аннотация. Ранее для каждой многослойной нейронной сети прямого распространения сигнала (далее нейронная сеть) вводились конечные коммутативные группоиды, которые получили название аддитивные группоиды подсетей. Данные группоиды тесно связаны с подсетями нейронной сети, над которыми они построены. Группоид является моноидом тогда и только тогда, когда он построен над двухслойной нейронной сетью. Ранее для данных группоидов изучались эндоморфизмы и их свойства, а также были построены некоторые эндоморфизмы, но исчерпывающего поэлементного описания не получено. Было показано, что всякий конечный моноид изоморфен некоторому подмоноиду моноида всех эндоморфизмов подходящего аддитивного группоида подсетей для некоторой подходящей нейронной сети. В работе рассмотрены эндоморфизмы аддитивных группоидов подсетей двухслойных нейронных сетей. Основным результатом исследования является поэлементное описание моноида всех эндоморфизмов аддитивных моноидов подсетей, построенных над двухслойной нейронной сетью. Поэлементное описание получено за счет построения общего вида эндоморфизма. Общий вид эндоморфизма параметризуется эндоморфизмами подходящих булеанов относительно операции объединения. Поэтому изучены эндоморфизмы данных булеанов, в том числе полукольца эндоморфизмов данных булеанов относительно объединения. Кроме того, для описания общего вида эндоморфизма аддитивного моноида подсетей использованы гомоморфизмы одного буалеана в другой (относительно объединения).

Ключевые слова: эндоморфизм группоида, многослойная нейронная сеть прямого распространения сигнала, подсеть многослойной нейронной сети

Благодарности: Работа выполнена при поддержке Красноярского математического центра и финансировании Министерства науки и высшего образования Российской Федерации (Проект № 075-02-2022-876).

Ссылка для цитирования: Литаврин А. В. Об эндоморфизмах аддитивного моноида подсетей двухслойной нейронной сети // Известия Иркутского государственного университета. Серия Математика. 2022. Т. 39. С. 111-126. https://doi.org/10.26516/1997-7670.2022.39.111

1. Introduction

This paper is a continuation of the study [4] in which the algebraic properties of some finite commutative groupoids AGS(^) are studied.

Groupoids AGS(.M") are built over a given multilayer neural network N with direct signal distribution. The elements of this groupoid model the subnets of the neural network M in the sense of Definition 4 from [4].

In [4], the groupoids AGS(.M") are called it additive subnet groupoids of multilayer neural network M.

Among the main problems considered in the work [4] the problem was of element-wise description of the monoid of all endomorphisms of the groupoid AGS(^). It was shown that every finite monoid can be isomor-phically embeddable into the monoid of all endomorphisms of the groupoid AGS(^) for a suitable neural network N. Some endomorphisms of the groupoid AGS(^) have been described, but an exhaustive description of the elements End(AGS(^")) was not received.

It turned out that the groupoid AGS(^) is a monoid if and only if M is a two-layer neural network (n(N) = 2). Moreover, if M is a two-layer neural network and M\ and M2 are the set of all neurons lying in the first and second layers, and B(X) := (2X, U), then the equality AGS(^) = B(M\) x B(M2) holds (equality of sets of supports and equality operations; a stronger condition than isomorphism).

The main result of this work is the element-wise description of the monoid of all endomorphisms End(AGS(^")), when n(N) = 2. To describe endomorphisms from End(AGS(^")) are used homomorphisms from B(X) to B(Y) and endomorphisms from End(5(X)) for special X. The paper considers ways of describing such homomorphisms and endomorphisms. It is well known that the set of all endomorphisms of a commutative monoid forms a semiring with respect to the standard addition of endomorphisms and the composition of endomorphisms. In this paper, a special matrix representation of the endomorphism semiring End(5(X)) is obtained for an arbitrary finite set X.

There are many studies on the properties of endomorphisms (in particular, automorphisms) of algebraic systems (see, for example, [9; 11; 12]). In particular, their element-wise descriptions. The properties of automorphisms of geometric objects are studied (see, for example, [8]).

Basic information about neural networks (in particular about multilayer neural networks) can be found in [2-5; 10]. It should be noted that the approach to determining the subnet of a multilayer neural network differs from the approach to determining the subsystem of a given algebraic system. In the theory of abstract automata (see, for example, the survey [1;6]), an abstract automaton is identified with a three-base algebraic system. The work [7] introduces the concept abstract neural network. This concept is similar to the concept of an abstract automaton, but differs in some specificity that is convenient for applying this abstraction to the study of issues specific to neural networks (in particular, training). There also arises the concept of an abstract neural network subnet, built as a subsystem of the

corresponding three-base algebraic system. This approach is fundamentally different from the approach of introducing the concept of subnet in [4].

It should be noted that, in essence, it is impossible to study the internal structure of a neural network from the standpoint of abstract automata, therefore, from the standpoint of abstract neural networks. This detail is well known and was noted by V.M. Glushkov in the review [1, p.59,conclusion].

2. Basic definitions related to neural networks

This section will define the notions of a multilayer neural network, its subnet and groupoid AGS(.M").

In this paper, sets will be denoted in capital Latin letters, and tuples composed of sets, in capital Latin letters with a bar. A tuple of empty sets will be denoted by the symbol 0 := (0,..., 0) (the length of such a tuple will always be clear from the context).

By default, R is the set of all real numbers. By F(R) we denote the set of all functions h : R ^ R (here it is understood that the domain of the function h coincides with the set R).

Next, we give definition 3 from [4].

Definition 1. Let the following objects be given:

1) a tuple (Mi,..., Mn) of length n > 1 of finite non-empty sets, where Mi П Mj = 0 is true for i = j;

2) the set S := (Mi x M2) U (M2 x M3) U ... U (Mn-1 x Mn);

3) the mapping f : S ^ R, which assigns a real number to each pair from S;

4) the set A := M1 U ... U Mn;

5) the mapping g : A ^ F(R), which assigns to each element from A a function from F (R);

6) the mapping I : A ^ R, which assigns to each element from A some number from R.

Then the tuple N = (M1,..., Mn, f, g, I) will be called a multilayer neural network of direct distribution (in the framework of this work, just neural networks).

The tuple (M1,..., Mn) is interpreted as the main tuple of neurons in the neural network N, S is interpreted as a set of synoptic connections. The f function defines the synoptic connection weights, and the g function defines the functions activation in each neuron. The I function defines the threshold values of neurons. The input layer will be called the set of neurons Mi.

Information about the standard operation of a neural network as a computational circuit can be found in [2-4] and others.

Let two tuples X = (X1, ...,Xn) and Y = (Yi,...,Yn) of finite nonempty sets be given. Then by X U Y we will denote the componentwise union_X U Y := (X1 U Y1, ...,_Xn U Yn).

If X = (X1, ...,Xn) and M = (Mi, ...,Mn) are two tuples whose components are sets, then we say that the condition X C M if all inclusions X1 C M1, ...,Xn C Mn are true (componentwise inclusion).

Let (X1, ...,Xn) be some tuple composed of finite sets, we say that the tuple is continuous if for all different i,j e {1,...,n} the following implication holds: if Xi = 0 and Xj = 0 and i < j, then for all s e {i, ...,j} the inequality Xs = 0 holds. The tuple 0 is assumed to be continuous by definition. For a tuple of sets to be continuous, it should not have alternation of a non-empty set with an interval of empty sets, and then again with a non-empty set.

Let us give definition 4 from [4].

Definition 2. Let the neural network be defined N = (M1,..., Mn, f,g, I) and a continuous tuple X = (X1, ...,Xn) is given such that it contains more than one component other than the empty set, and

(Xu..,Xn) C (M1,..,Mn).

We assume that Y = (Y1, ...,Ym) is a tuple obtained from a tuple X by deleting components equal to the empty set, where m < n. If f' is a restriction of the function f on the set

S' := (Y1 x Y2) U (Y2 x Y3) U ... U (Ym-1 x Ym)

and g', I' is the restriction of the functions g and I on the set N := Y1 U ... U Ym, then object

N' :=(Y1 ,...,YmJ',g',l')

will be called subnet of the network N. We say that the tuple X induces the subnet N'. The Y tuple is the main tuple of neurons in the N' subnet. In general, the tuples X and Y can be different.

More information about neural network subnets can be found in [4]. Note that the proposed approach to defining the subnetwork of a neural network corresponds to works studying the applied aspects of neural networks.

Construction of groupoids AGS(^). Next, we formulate Definition 1 from [4] groupoid AGS(^").

Definition 3. Let the neural network N be defined with the main tuple of neurons M. The set of all possible continuous tuples X C M will be denoted by the symbol AGS(^).

We assume that X and Y are two arbitrary element from AGS(^). Let's define a binary algebraic operation (+):

X U Y, ifX U Y e AGS(^) 0, ifX U Ye AGS(^).

Then the groupoid AGS(^) := (AGS(^), +) will be called the additive groupoid of subnets of neural network N.

If M is a two-layer neural network, then for all X,Y e AGS(^) the equality holds X + Y = X U Y and AGS(^) = B(Mi) x B(M2) (equality of sets).

X + Y : =

3. Some definitions and formulation of the main result

Let us formulate the necessary definitions. Let G = (G, o) be a monoid and 1 £ G is a neutral element of this monoid. Then the mapping ф : G ^ G is called an endomorphism of the monoid G if = 1 and for all x,y £ G the equality is true

(X o у)ф = хф o y^. (3.1)

A semiring is a non-empty set S with two binary algebraic operations (+) and (*) such that (S, +) is a commutative monoid, (S, ■) is a semigroup, addition and multiplication are related by left and right distributivity with respect to addition, and a neutral element о of the monoid (S, +) satisfies the identity о ■ x = x ■ о = о (multiplicative property of zero). It is well known that the set of all endomorphisms of a commutative monoid with respect to the standard operation of addition of two endomorphisms and the composition of two endomorphisms forms a semiring. Let G be a commutative monoid. Notation related to composition of endomorphisms. We assume that x £ G and ф £ End(G). Then x^ is the image of the element x under the action of the endomorphism ф. The composition of two endomorphisms will be denoted by the symbol (■). If ф1,ф2 £ End(G) and ж £ G, then хф1^2 := (хф2)ф1.

Notation related to the sum of endomorphisms. Let be ф1,ф2 £ End(G) and x £ G. Then, as usual, the sum (+) of two endomorphisms will denote the mapping ф1 + ф2, which acts on G according to the rule

:= I

Ш : - Ш | Ш .

It is well known that the sum of two endomorphisms of a commutative monoid is again an endomorphism of this monoid.

Let X be some finite set, 2X a Boolean of the set X. We will use the notation В(X) = (2х, U). In the framework of this paper, we consider the

Boolean of some set only with respect to the operation (U). It is well known that B(X) = (2X, U) is a commutative monoid consisting of idempotents.

We assume that N is a two-layer neural network with the main tuple of neurons (M1 ,M2). As noted in the introduction, the equality is true AGS(^) = B(M1) x B(M2).

For any endomorphism t1 e End(B(M1)) and any homomorphism t2 of the monoid B(M2) into the monoid B(M1) we introduce the mapping

aT1,T2 (U) = Ul1 U Ul2 (U = (U1, U2) e AGS(^)).

The mapping aT1,T2 is a homomorphism from AGS(.M")) = B(M\) x B(M2) to B(Mi). Indeed, let U = (^1,^2) and V = (Vi,V2) be two arbitrary elements from AGS(^)). We have the equalities

an,t2 (U U V) = (Ui U Vi)T1 U (U2 U V2)T2 = Ul1 U V^1 U UT22 U Vp =

[Ul1 U UT2] U [V? U V?] = an>T2(u) U an>T2 (V).

Thus, we have shown that aT1,T2 is a homomorphism.

For every homomorphism £1 of the monoid B(M1) into the monoid B(M2) and every endomorphism £2 e End(B(M2)) we introduce the mapping _ _

Pcuc2(u) = Ul1 U U22 (U = (U1, U2) e AGS(^)). The mapping 22 is a homomorphism of the monoid B(M1) x B(M2) into B(M2). Indeed, let U = (U1, U2) and V = (V1, V2) be two arbitrary elements from AGS(^)). We have the equalities

fcuc2 (U U v) = (U1 U V1)21 U (U2 U )22 = U21 U V-21 U u22 U v22 =

[U21 U U22] U [V21 U ^] = $(1,22 (U) U ,%,22 (V). Thus, we have shown that ^,22 is a homomorphism.

For any t\ e End(5(Mi)), (2 e End(B(M2)), arbitrary homomorphisms t2 of the monoid B(M2) into the monoid B(M\) and of the monoid B(Mi) into the monoid B(M2) we introduce the mapping p : B(M\) x B(M2) ^ B(Mi) x B(M2) given by the rule

UP = (an,T2(U),Pc1,22(U)) (U e B(X) x B(X)). (3.2)

Let us show that the mapping p introduced by the rule (3.2) is an endomorphism of the monoid AGS(^). Let U = (Ui,U2) and V = (Vi,V2) -two arbitrary elements from AGS(^). We get equalities

(U + V) = (aT1,T2 (U + V),f321,22 (U + V)) = (a^ (U U V),^ c2 (U U V)) =

(an,T2(U) U an,T2(V),P21,22(U) U P21,22(V)) = Up + The main theorem in this work is the theorem

Theorem 1. The set of all endomorphisms of the monoid AGS(^) for n(N) = 2 is bounded by all kinds of endomorphisms p.

Thus, an arbitrary endomorphism of the monoid AGS(^) is parameterized by homomorphisms from AGS(.W) (n(M) = 2) to B(M1), B(M2). These homomorphisms are parameterized by homomorphisms (in particular, endomorphisms) from B(X) to B(Y), when X = M1,M2 and Y = M1,M2. Therefore, in this paper we prove Proposition 1 (see the next section).

Proposition 1 gives an element-wise description of all homomorphisms of the Boolean B(X) into B(A). A consequence of Proposition 1 (see Corollary 1, next section) is an element-wise description of all endomorphisms of the Boolean B(X) for an arbitrary finite set X.

For the monoid of all endomorphisms End(5(X)) for an arbitrary finite set X one can establish a matrix representation over a special semiring. As noted above, End(5(X)) is a semiring under addition and composition of two endomorphisms.

Next, we need basic binary logic functions: conjunction (we will denote (A)) and disjunction (we will denote (V)). We will use the logical semiring B = ({0,1}, V, A). The set of all possible square matrices of order n with elements from the ring B will be denoted by Mnxn(B).

Theorem 2. For each finite set X consisting of n elements, the semiring End(5(X)) of all endomorphisms of the monoid B(X) = (B(X), U) is isomorphic to the semiring of matrices Mnxn(B) with elements from the logical semiring B.

4. Homomorphisms from В (A) to B(C)

Consider homomorphisms from the Boolean B(A) to the Boolean В (С). It is easy to show that the set Q = {0} U {{ж} | x £ A} is a generating set of the monoid В (A).

General view of the homomorphism from B(A) to B(C). For each family С = {Lx}xeA of sets from 2C define the mapping фс given by the rule

ифс = У Lu, 0ф£- = 0 ueu

for any non-empty set U £ B(A). Since the inclusions Lx £ B(C) and U £ В (A) holds, then the inclusion ифс £ В (С) holds.

Lemma 1. The mapping фс is a homomorphism of the monoid B(A) = (2a, U) into the monoid B(C) = (2C, U).

Proof. Next, 0 := <pc. Let U and V be two arbitrary elements from B(A). Since the monoid B(A) is commutative, associative, and idempotent, the equality is true

(U U V)+ = U Lm = ( U ¿J u( U lA .

meuuv \ueu J \vev J

On the other hand, the equality is true U^ U V^ = (LLeu U (UveV Lv). Thus, we have shown that for any U,V G B(A) the equality is true

(U U V)+ = U+ U V

The lemmae is proved. □

Obviously, if equality A = C is true and the family £ = {Lx}xEa of sets from 2C is defined, then ^c is an endomorphism of the monoid B(A).

Proposition 1. Any homomorphism of the monoid B(A) into the monoid B(C) is a homomorphism for a suitable family £ of subsets from B(C).

Proof. Let 0 be an arbitrary monoid homomorphism B(A) to B(C).

1. It is clear that 0^ = 0. Consider the action 0 on the generating set Q. We assume that the image of the element {x},x G A under the action of 0 is equal to the set Wx g B(C).

We introduce a family £ = {Lx} x^a of sets Lx from B(C) such that for all x G A the equality is true Lx = Wx.

The family £ is defined so that for all x G A the equalities are true

{X}^ = Lx = WX = {x}* ({x}G Q).

Thus, we have shown that every homomorphism $ acts on the set Q as a homomorphism

It remains for us to show that 0 acts on B(A) \ Q as endomorphism 2. Suppose that U G B(A) \ Q. For this element, the decomposition U = Uxe u{%} is valid and the equalities are true

c

u* =( U to) = U to* = U to*c = ( U to) = u*

\xeu J xeu xeu \xeu )

Thus, an arbitrary homomorphism 0 acts on B(A) as a homomorphism ^c for a suitable family £. □

If we assume in Proposition 1 (and its proof) that A = C = X, then we obtain

Corollary 1. Any endomorphism of the monoid B(X) is an endomorphism for a suitable family £ = [Lx]xex of subsets from B(X).

5. Composition and sum of two endomorphisms B(X)

Let there be given two families of sets £ = {Lx}xeX, ^ = {Dx}xeX from 2x. The mappings <fic and are endomorphisms of B(X). Then the equalities are true

• = , <pc + = ,

where family members Z = {Zx}xeX and V = {Vx}xeX satisfy the equalities

z I \JyeDx Lv> if — 0

(5.1)

if Dx = 0, Vx = Lx U Dx. (5.2)

Indeed, let U be an arbitrary element from 2X. Then the equalities hold

-4>v = (u^v= i y ^ = y D= U f U L

\xeu J xeu xeu \yeDx

Zx :— (^J Ly

yeDx

U — U^

xeu

where Z = {Zx}xeX. Equality (5.1) is proved. On the other hand, the equalities are true

u^= u^ U = ( U ¿J U ( U D") = U (Lu U Du) =

\ueu J \ueu J ueu [Fx := Lx U Dx] = U K = U^.

ueu

Equality (5.2) is proved.

6. Proof of Theorem 2

Further, we need the basic binary logical functions: conjunction (we will denote (Л)) and disjunction (we will denote (V)). We will use the logical semiring В = ({0,1}, V, Л).

Further, we assume that the set X is finite (|X| = n) and ordered. In accordance with this ordering, we will denote the elements of the set X = {xi, ...,xn}.

Let a family of sets С = {Lx}xex be given. Then the endomorphism фс is defined. Since the order is defined on the elements of the set X, specifying

the family £ is equivalent to specifying the tuple L = (L\,..., Ln), where Li := Lxh . _

For each endomorphism , where L = (L\, ...,Ln) we define a square matrix A— with elements from B of order n as follows:

1) al = (an);

2) ai:?- = 1 if and only if xi G Lj;

3) ai:?- = 0 if and only if ^ G Lj.

This way of assignment can be reformulated in words. If the component Lj contains the element xi, then the element in the j-th column and in the i-th row is equal to one, otherwise zero. Thus, the matrix A— is compiled column by column.

The set of all possible square matrices of order n with elements from the ring B will be denoted by Mnxn(B).

Next, consider the mapping a : End(5(X)) ^ Mnxn(B) defined by rule

a(<h) = AlFrom the way of constructing the matrix A— it can be seen that the mapping a is a bijection. Let us show that a is an isomorphism between the semirings End(5(X)) and Mnxn(B).

To show that a is an isomorphism, we show that for any tuples L = (Li,..., Ln) and D = (Di,..., Dn) from (2X)n equalities are true

a(0z •0v)=A— • %, (6.1)

a(<h + 0d)=Al + Ad , (6.2)

where on the right stand the usual matrix multiplication and matrix addition.

Isomorphism of multiplicative semigroups of semirings. Next, we will show that the equality (6.1). Let be

:= • 0d' where, by virtue of (5.1), the equalities

= [Ux^Li, ifD = 0 3 \0, if Dj = 0.

We have the equality a(<frz) = We assume that A-% = (z^), A— = (aij), Ad = (bij) and A— • Ad = C = (dj) where

V (aik A bkj).

k=i

Let dj = 1. This means that there are elements aik' and bk'j equal to one. Which in turn means that Dj contains the element Xk<, and the set

i =

Lk> contains the element xi. Hence, the set Zj contains the element xi, therefore, the matrix A-% contains the element zi j = 1 = ci j.

Let Cij = 0. This is possible in one (and only one) of the cases:

1. all bk j are equal to zero for any 1 < k < n;

2. among the elements of bkj there are nonzero elements, denote them

by

{bk1,j,bk2,j, ...,bks,j},

but all elements {ai,kl ,ai,k2, ...,ai,ks} are equal to zero.

In the first case, we get that the set Dj is empty, therefore, the set Zj is also empty. Hence, zi j = 0 = ci j (in this case, for any i). In the second case, we get that the set Dj contains elements {^kl, xk2 ,...,xks} and sets {Lkl,Lk2,...,Lks} do not contain element xi, therefore, the following conditions are true

h= s

Zj—U Lkh >

h=l

This means that Zij = 0 = Cij (in this case, for specific i and j).

Thus, we have shown that the matrices A-- = (z^) and A— • A— = C are equal. Since • ) = a(<fi—) = A-- = A— • A— , then the identity (6.1) also holds.

Isomorphism of additive commutative monoids of semirings. Let be

:= ^L + ^D,

where by virtue of (5.2) the relations Vj = Lj U Dj.

We have the equality a(^y) = Ay. We assume that Ay = (v^) and A- + A-— = W = (wy) where w^ = a^ V b^.

Let be w-ij = 1. Hence, a^ or b^ equal to one. Means what x, G Lj or x, G Dj, hence, x, G Lj U Dj = Vj. Hence we obtain the equality

Wi j = Vi j = 1.

Let be Wj,j = 0. Therefore, a^ = 0 and b^ = 0. Hence, x, G Lj and Xj, G Dj, hence, x, G Lj U Dj = Vj. Hence we obtain the equality

Wij = Vjj = 0.

Thus, we have shown that the matrices Ay and W = A- + A— are equal. Therefore, the equality (6.2) is also true. Thus, we have proved Theorem 2.

7. Proof of the main theorem 1

1. Let ф be an endomorphism of the monoid AGS(^) for n(M) = 2 and U from AGS(^). Then the equalities are true

йф = (Ri(U ),R2(U)),

where Ri : AGS(^) ^ B(Mi) and R2 : AGS(^) ^ B(M2). It is trivially established that Ri is a homomorphism from AGS(^) to B(Mi) and R2 is a homomorphism from AGS(^) to B(M2). Indeed, since n(M) = 2, then U + V = U U V and for arbitrary U and V equalities hold

Thus, the equalities are true

Ri(U + V) = Ri(U) U Ri(V), R2(U + V) = R2(U) U R2(V).

Next, we need a description of all endomorphisms from AGS(.M") to B(Mi) and B(M2).

2. Let us show that all homomorphisms from AGS(^) to B(Mi) are exhausted by the homomorphisms aTl,T2.

Let p be an arbitrary homomorphism from AGS(^) to B(Mi). The monoid AGS(.W) will be generated by the set Ti UTr U {(0,0)}, where

T := {(0, {£}) | z G M2}, Tr := {(M, 0) | x G Mi}.

Let Ux := (0, {c}) G Ti and Vy := ({y}, 0) G Tr. Then the equalities hold

p(Ux) =Hx G B(Mi), p(Vy) = By G B(Mi), p((0, 0)) = 0.

We define two families of sets

such that Lx = Hx and Dy = By. Next, consider the endomorphism n = and the homomorphism r2 = It is easy to show that aT1,T2 ((0,0)) = 0. Mapping aT1,T2 will satisfy the equalities

P(Ux) = «T1,T2 (Ux), P(Vy) = «T1,T2 (Vy), p((0, 0)) = «T1,T2 ((0, 0))

for all X G M2, y_G Mi.

Further, let U = (U, V) be an arbitrary element from B(Mi) x B(M2). Consider the action p on this element (the tuples Ux and Vy are defined above)

(U + V)* = (Ri(U + V), R2(U + V));

(U + V)+ = U* + = (Ri(U), R2(U)) + (Ri(V), R2(V)) (Ri(U) U Ri(V), R2U U R2(V)).

£ = {Lx}x£m2

C B(Mi), V = {Dx}x£Mi C B(Mi)

P(U) = p P(Vy)

U aTuT2(Ux) U [J aTl,T2 (Vy) = aTl,T2 I U Ux u U Vy I = «n,r2 (U). xeu yev \xeu yev J

Thus, we have shown that on an arbitrary element U from AGS(^), the homomorphism <p acts as a homomorphism aT1,T2.

Similarly, one can show that every homomorphism ф from AGS(.M") to B(M2) acts as a homomorphism ß^1,^.

3. Thus, the homomorphism R\ is a suitable homomorphism aT1,T2, and the homomorphism R2 is a suitable homomorphism ß^1,ç2. And an arbitrary endomorphism of the monoid AGS(^) is an endomorphism of p given by the rule (3.2). Theorem 1 is proved.

8. Conclusion

Thus, an arbitrary endomorphism of the monoid AGS(^) is parameterized by homomorphisms from AGS(.W) (n(M) = 2) to B(Mi), В(M2). These homomorphisms are parameterized by homomorphisms (in particular, endomorphisms) from B(X) to B(Y), when X = Mi,M2 and Y = Mi, M2. Proposition 1 and its Corollary 1 give an element-wise description of these homomorphisms and endomorphisms. And Theorem 2 gives more detailed information about the structure of the semiring End(X), for an arbitrary finite set X.

These results are of theoretical and practical interest. These results can be used to carry out calculations in the construction or theoretical study of multilayer neural networks of direct signal propagation.

References

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Kravtsova O.V. Elementary Abelian 2-subgroups in an Autotopism Group of a Semifield Projective Plane. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 32, pp. 49-63. doi.org/10.26516/1997-7670.2020.32.49 Litavrin A.V. Endomorphisms of some groupoids of order k + k2. The Bulletin of the Irkutsk State University. Series Mathematics, 2020, vol. 32, pp. 64-78. https://doi.org/10.26516/1997-7670.2020.32.64

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Tsarkov O.I. Endomorphisms of the semigroup G2(r) over partially ordered commutative rings without zero divisors and with 1/2. J. Math. Sc., 2014, vol. 201, no. 4, pp. 534-551.

Zhuchok Yu.V. Endomorphism semigroups of some free products. J. Math. Sci., 2012. vol. 187, no. 2, pp. 146-152.

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Литаврин А. В. Эндоморфизмы конечных коммутативных группоидов, связанных с многослойными нейронными сетями прямого распределения // Труды ИММ УрО РАН. 2021. Т. 27, № 1, С. 130-145. https://doi.org/10.21538/0134-4889-2021-27-1-130-145

Литинский Л. Б. О задаче декомпозиции нейронной сети на несколько подсетей // Математическое моделирование. 1996. Т. 8, № 11. С. 119-127. Плоткин Б. И., Гринглаз Л. Я., Гварамия А. А. Элементы алгебраической теории автоматов. М. : Высшая школа, 1994. 192 с.

Слеповичев И. И. Алгебраические свойства абстрактных нейронных сетей // Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2016. Т. 16, № 1. С. 96-103. https://doi.org/10.18500/1816-9791-2016-16-1-96-103

Kravtsova O.V. Elementary Abelian 2-subgroups in an Autotopism Group of a Semifield Projective Plane // The Bulletin of the Irkutsk State University. Series Mathematics. 2020. Vol. 32. P. 49-63. doi.org/10.26516/1997-7670.2020.32.49 Litavrin A.V. Endomorphisms of some groupoids of order к + к2 // TheBulletin of the Irkutsk State University. Series Mathematics. 2020. Vol. 32. P. 64-78. doi.org/10.26516/1997-7670.2020.32.64

McCulloh W., Pitts W. A logical calculus of the ideas immanent in nervous activity // Bulletin Math. Biophysics. 1943. N 5. P. 115-133.

Tsarkov O.I. Endomorphisms of the semigroup G2(r) over partially ordered commutative rings without zero divisors and with 1/2 // J. Math. Sci. 2014. Vol. 201, N 4. P. 534-551.

Zhuchok Yu.V. Endomorphism semigroups of some free products //J. Math. Sc. 2012. Vol. 187, N 2. P. 146-152.

Об авторах

Литаврин Андрей Викторович,

канд. физ.-мат. наук, доц., Сибирский федеральный университет, Российская Федерация, 660041, г. Красноярск, [email protected], https://orcid.org/0000-0001-6285-0201

About the authors

Andrey V. Litavrin, Cand. Sci. (Phys.-Math.), Assoc. Prof., Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, [email protected], https://orcid.org/0000-0001-6285-0201

Поступила в 'редакцию / Received 13.12.2021 Поступила после рецензирования / Revised 19.01.2022 Принята к публикации / Accepted 27.01.2022