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Известия Саратовского университета. Новая серия. Серия: Математика
Механика. Информатика. 2024. Т. 24, вып. 3. С. 320-329
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2024:
vol. 24, iss. 3, pp. 320-329
https://mmi.sgu.ru
https://doi.org/10.18500/1816-9791-2024-24-3-320-329 EDN: DGQYFY
Article
On semigroups of relations with the operation of the rectangular product
D. A. Bredikhin
Saratov State University, 83 Astrakhanskaya St., Saratov 410012, Russia Dmitry A. Bredikhin, i.a.bredikhin@mail.ru, https://orcid.org/0000-0002-5155-8499, AuthorlD: 2925
Abstract. A set of binary relations closed with respect to some collection of operations on relations forms an algebra called an algebra of relations The theory of algebras of relations is an essential part of modern algebraic logic and has numerous applications in semigroup theory. The following problems naturally arise when classes of algebras of relation are considered: find a system of axioms for these classes, and find a basis of of identities (quasi-identities) for the varieties (quasi-varieties) generated by these classes. In the paper, these problems are solved for the class of semigroups of relation with the binary associative operation of the rectangular product, the result of which is the Cartesian product of the first projection of the first relation on the second projection of the second one.
Keywords: algebra of relations, primitive positive operation, variety quasi-variety, semigroup, partially ordered semigroup For citation: Bredikhin D. A. On semigroups of relations with the operation of the rectangular product. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2024, vol. 24, iss. 3: pp. 320-329. https://doi.org/10.18500/1816-9791-2024-24-3-320-329 EDN: DGQYFY
This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0)
Научная статья УДК 501.1
О полугруппах отношений с операцией прямоугольного произведения
Д. А. Бредихин
Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского, Россия, 410012, г. Саратов, ул. Астраханская, д. 83
Бредихин Дмитрий Александрович, доктор физико-математических наук, профессор кафедры геометрии, i.a.bredikhin@mail.ru, https://orcid.org/0000-0002-5155-8499, AuthorID: 2925
Аннотация. Множество бинарных отношений, замкнутое относительно некоторой совокупности операций над отношениями, образует алгебру, называемую алгеброй отношений. Теория алгебры отношений является существенной частью современной алгебраической логики и имеет многочисленные приложения в теории полугрупп. При рассмотрении классов алгебры отношений естественно возникают следующие проблемы: найти систему аксиом для этих классов, найти базис тождеств (квазитождеств) для многообразий (квазимногообразий), порожденных этими классами. В статье обозначенные проблемы решаются для класса полугрупп отношений с бинарной ассоциативной операцией прямоугольного произведения, результатом которой является декартово произведение первой проекции первого отношения на вторую проекцию второго.
Ключевые слова: алгебра отношений, примитивно-позитивная операция, многообразие, квазимногообразие, полугруппа, частично упорядоченная полугруппа
Для цитирования: Bredikhin D. A. On semigroops of relations with the operation of the rectangular product [Бредихин Д. А. О полугруппах отношений с операцией прямоугольного произведения] // Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2024. Т. 24, вып. 3. С. 320-329. https://doi.org/10.18500/1816-9791-2024-24-3-320-329, EDN: DGQYFY Статья опубликована на условиях лицензии Creative Commons Attribution 4.0 International (CC-BY 4.0)
Introduction
Let Rel(U) be the set of all binary relations on a base set U. A set of binary relations $ c Rel(U) closed with respect to some collection Q of operations on relations forms an algebra
Q) called an algebra of relations. The theory of algebras of relations is an essential part of modern algebraic logic and has numerous applications in the theory of semigroups (see the remarkable survey [1]).
Denote by R{Q} the class of all algebras isomorphic to the ones whose elements are binary relations and whose operations are members of Q. Let V{Q} be the variety and let Q{Q} be the quasi-variety generated by R{Q}.
The following problems naturally arise when the class R{Q} is considered.
1. Find a system of axioms for the class R{Q}.
2. Is the class R{Q} finitely axiomatizable?
3. Find a basis of quasi-identities for the quasi-variety Q{Q}.
4. Is the quasi-variety Q{Q} finitely based?
4. Find a basis of identities for the variety V{Q}.
6. Is the variety V{Q} finitely based?
7. Does the class R{Q} form a quasi-variety?
8. Does the quasi-variety Q{Q} form a variety?
Numerous studies have been devoted to solving these problems for various classes of algebras of relations. The first mathematician who treated algebras of relations from the point of view of universal algebra was A. Tarski [2]. He considered algebras of relations (Tarski's algebras of relations) with the following operations: Boolean operations U, n, - ; operations of relational product o and relational inverse -1; constant operations A (diagonal relation), 0 (empty relation), V = U x U (universal relation). He showed that the class R{o, -1, U, n, -, A, 0, V} is not a quasi-variety and the quasi-variety generated by this class forms a variety [3]. R. Lyndon [4] found the infinite base of this variety and D. Monk [5] showed that it is not finitely based.
A little later, B. Jonsson considered the class R{o, -1, n, A}, proved that it forms a quasi-variety, found its infinite base of quasi-identities, and posed problems 4 and 8 for this class [6]. The negative solutions to these problems were obtained in [7] and [8] respectively.
Operations on relations are usually determined using first-order predicate calculus formulas.
Such operations are called logical. It is known that classes of algebras of relations with logical operations are axiomatizable [1]. One of the most important classes of logical operations on relations is the class of primitive-positive operations [9] (in other terminology — Diophantine operations [10]). An operation on relations is called primitive-positive if it can be defined by a formula of the first-order predicate calculus containing in its prenex normal form only existential quantifiers and conjunctions. Note that the set-theoretical inclusion c is compatible with all primitive-positive operations. Thus, any algebra of relations with primitive-positive operations Q) can be considered as partially ordered Q, c). The corresponding abstract class of partially ordered algebras will be denoted by R{Q, c}. The variety and the quasi-variety generated by the class R{Q, c} will be denoted by V{Q, c} and Q{Q, c} respectively. Problems 1-8 for the class R{Q, c} are formulated in the same way.
One of the most important associative primitive-positive operations is the operation of relational product o that is defined as follows:
It is well known that the class R{o} coincides with the class of all semigroups and the class R{o, c} coincides with the class of all partially ordered semigroups. There are many other binary primitive-positive operations on relations (see, for example [11]). It is interesting to consider problems 1-8 for algebras of relations with these operations. The paper provides a solution to these problems for the class of semigroups of relations with the operation of the rectangular product.
1. Main results
We concentrate our attention on the following primitive-positive operation:
It is easy to see that this operation is associative. Note also that p * a = pr1 p x pr2a, where = {u : (3£)(u, t) e p} is the first projection of p and pr2a = {v : (3w)(w,v) e a} is the second projection of a. Since pr1 p x pr2a is a rectangular relation, we will treat this operation as the operation of the rectangular product.
A partially ordered semigroup is an algebraic system (A, •, <), where (A, •) is a semigroup and < is a partial order relation on A that is compatible with multiplication, i.e., x < y implies xz < yz and zx < zy for all x, y, z e A.
The main results are formulated in the following theorem and corollaries. Their proofs are based on the description of quasi-equational theories of algebras of relations with primitive-positive operations [10].
Theorem. The quasi-variety Q{*, c} forms a variety in the class of all partially ordered semigroups. A partially ordered semigroup (A, •, <) belongs to the quasi-variety Q{*, c} if and only if it satisfies the identities:
Corollary 1. The quasi-variety Q{*} forms a variety. A semigroup (A, •) belongs to the quasi-variety Q{*} if and only if it satisfies the identities:
p о a = {(u, v) : (3 t) G p Л (t, v) G a}.
p * a = {(u, v) : (3t, w)(u, t) G p Л (w, v) G a}.
x < x2,
(1) (2)
xyz < xz.
(xy)2 = xy,
(3)
(4)
(5)
xyz = xyxz,
xyzx = xzyx.
Corollary 2. The class R{*, C} does not form a quasi-variety. For a partially ordered semigroup (A, •, <), the following three conditions are equivalent.
1. (A, •, <) belongs to the class R{*, C}.
2. One of the following conditions holds:
a) (A, •, <) satisfies identity (1) and the identity
xyz = xz; (6)
b) (A, •, <) contains the zero element o, satisfies identity (1) and the axioms:
y = o ^ xyz = xz, (7)
o < x. (8)
3. (A, •, <) satisfies identity (1) and the axioms:
xyz = xy V yw = wy = y, (9)
xy = yx = x ^ x < z. (10)
Corollary 3. The class R{*} does not form a quasi-variety. For a semigroup (A, •) the following three conditions are equivalent.
1. (A, •) belongs to the class R{*}.
2. One of the following conditions holds:
a) (A, •) satisfies identity (6);
b) (A, •) contains the zero element o and satisfies axiom (7).
3. (A, •) satisfies axiom (9).
Note also that if the semigroup ($, o) of rectangular relations does not contain a zero element, then the operations o and * coincide. It follows that the equivalence conditions 1 and 2a of Corollary 3 can be obtained from the results of [12]. Note that the next problem is still open.
Problem. Let (A, •, <) [respectively, (A, •)] be a partially ordered semigroup [respectively, a semigroup] such that the conditions of Corollary 2 [respectively, Corollary 3] hold, and the set A is finite. Is it possible to isomorphically represent (A, •, <) [respectively, (A, •)] as a partially ordered semigroup ($, *, C) [respectively, as a semigroup ($, *)] of relations on an appropriate finite set U.
2. Proofs of results
Step 1. Let us consider the relationship between considered identities and axioms.
Lemma 1. Identities (1), (2) imply identities (3)-(5). Identity (6) implies identities (2)-(5). Identities (3)-(5) imply the following three identities:
xyzt = xzyt, (11)
xy = xy2, (12)
xy = x2 y. (13)
Proof. First of all, we show that identities (1), (2) imply identities (3)-(5). Indeed,
xy < q(xy)2 and (xy)2 = xyxy < xy, i.e., (xy)2 = xy (3). Further, xyxz < xyz and (1) 2 (2) (1) 2 (2) xyz < (xyz)2 = xyzxyz < xyxz, i.e., xyz = xyxz (4). Since xyzx < (xyzx)2 = xyzxxyzx < (1) 2 (2) < xzyx and xzyx < (xzyx)2 = xzyxxzyx < xyzx, we have xyzx = xzyx (5). It is clear
that identity (6) implies identities (2)-(5). Let us show that identities (3)-(5) imply identities
(4) ________, (5) ________, (4) w v (3) , 2 _ (4) _ 2
(11)—(13). Indeed, xyzt = xyzxt = xzyxt = xzyt (11); xy = (xy)2 = xyxy = xyy = xy2 (12);
(3b a2 (11) 2 2 (12) 2 , ч m
xy = (xy)2 = xyxy = xxyy = x2y2 = x2y (13). U
Lemma 2. Conditions 2 and 3 of Corollaries 2 and 3 are equivalent.
Proof. Note that we can represent axiom (6) as (-(Vw) yw = wy = y) ^ xyz = xz. It follows that this axiom is equivalent to identity (6), if A does not contain a zero element, and it is equivalent to axiom (7) otherwise. Axiom (10) just expresses axiom (8) as a universal formula of the first order language. □
Let us consider the set A = {x1,..., xn,... } of individual variables that are interpreted as elements of a semigroup. A semigroup term p is a word in the alphabet A, i.e., an expression of the form xj1xj2.. .xjm-1 xjm. For convenience, we will also use other letters of the Latin alphabet as variables.
Suppose that p = xj1 xj2 ...xjm-1 xjm be the term of a semigroup that satisfies identities (11)-(13). Then without loss of generality, we can assume that all variables xj2,...,xjm-1 are different and different from variables xj1 ,xjm. Moreover, we can also presume that variables xj2,... ,xjm-1 can be arbitrarily permuted. In what follows, we will use these properties without special mentions.
Step 2. Let ($, *, c) be the partially ordered semigroup of relations with the operation of the rectangular product. Since p * a = pr1 p x pr2a, we have p c pr1 p x pr2p = p * p, i.e., identity (1) holds. Note that p * n * a = pr1 p x pr2a = p * a, if n = 0, and p * n * a = 0 otherwise. It follows that identity (2) holds. It also follows that if 0 e then ($, *) satisfies identity (6). If 0 e then 0 is a zero element and axioms (7) and (8) hold. Thus, according to Lemmas 1 and 2, we obtain that all conditions of Theorem and Corollaries 1-3 are necessary.
Further, it is easy to see that for U = 0, the Cartesian square of the semigroup (Rel(U), *) of relations contains the zero element (0, 0) and does not satisfy axiom (7). It follows that the classes R{*} and R{*, c} do not form quasi-varieties.
Step 3. The proof of the sufficiency of conditions of the Theorem is based on the result of [10]. Let us give some definitions and notations to formulate this result. For any formula ^(z0,z1 ,r1 ,...,rm) of the first-order predicate calculus having m binary predicate symbols r1,... ,rm and two free individual variables z0, z1, we can associate an m-ary operation F^ on Rel(U) defined in the following way:
F^(p1,..., pm) = {(u, v) e U x U : <p(u, v, p1,..., pm)},
where ^(u, v, p1,... ,pm) means that the formula ^ holds whenever z0, z1 are interpreted as u, v, and r1,... ,rm are interpreted as relations p1,... ,pm from Rel(U). Recall that an operation on relations is called primitive-positive if it can be defined by a first-order formula containing in its prenex normal form only existential quantifiers and conjunctions. Let us describ primitive-positive operations by using graphs [9].
Let N be the set of all natural numbers and N0 = N U {0}. A labeled graph is a pair G = (V(G), E(G)), where V(G) is a finite set, called a vertex set, and E(G) c V(G) x N x V(G)
is a ternary relation. A triple (u,k,v) e E(G) is called an edge from u to v labeled by k, and it
k
will be graphically represented by u- ^ •v. An input-output-pointed labeled graph is a structure G = (V(G), E(G), in(G), out(G)), where (V(G),E(G)) is a labeled graph, in(G) and out(G) are two distinguished vertices (not necessarily different) called input and output vertices respectively. The input-output-pointed labeled graph G with in(G) = i and out(G) = o is also denoted by G%,°. In what follows, we shall usually speak simply of graphs if it does not lead to confusion. The concept of graph isomorphism is defined in a natural way. All graphs will be considered up to isomorphism.
For given u e V(G), the number of edges of the form (u, k,v) [respectively, (v,k,u)] we denote by deg+ (u) [respectively, deg-(u)].
Given two input-output-pointed labeled graphs G1 = (V1,E1, in1, out1) and G2 = (V2,E2, in2, out2), a mapping f: V2 ^ V1 is called a homomorphism from G2 to G1 if f (in2) = in1, f (out2) = out1, and (f (u), k, f (v)) e E1 whenever (u, k, v) e E2. We write G1 ^ G2 if there exists a homomorphism from G2 to G1.
Let F = F^ be a primitive positive operation determined by a formula Then the input-output-pointed labeled graph G = GF = G^ associated with F is defined as follows (see [5]). Let {0,1,..., n} be the set of all subscripts of individual variables of Put G = (V, E, in, out), where V = {vo, vi, ...,vn}; in = vo, out = vi; (i, k,j) G E if and only if the atomic formula rk(z,zj) occurs in
Note that the graph G* = (V, E, in, out) corresponding to the considered operation * of the rectangular product can be described in the following way:
V = {vo,vi,v2,v3}, E = {(vo, 1,v3), (v2, 2, vi)}, in = vo, out = vi, in = v0• —y -v3 v2• —y •vi = out.
Let G = (V, E, in, out) and Gk = (Vk, Ek, ink, outk) (k = 1,2,..., n) be graphs with pairwise disjoint vertex sets. The composition G(Gi,G2,..., Gn) is the graph constructed as follows [5]: take G and substitute every edge (u, k, v) G E by the graph Gk identifying the input vertex ink with u and the output vertex outk with v.
For any semigroup term p define the graph G(p) = (V(p),E(p), in(p), out(p)) in the following inductive way:
1) if p = xk, then G(p) is the following graph: in- A •out;
2) if p = piP2, then G(p) = G*(G(pi),G(p2)).
According to the construction, for any term p = xj1 xj2... xjm-1 xjm the graph G(p) has the following form:
in = v0• —> • • —>.....—> • • —> •vi = out.
Let G be a labeled graph, u, v G V(G) = {v0, vi,..., vn}, and Q be an input-output-pointed labeled graph. Without loss of generality, we can suppose that V(Q) = {wo, wi, ...,wm}, in(Q) = wo = u, out(Q) = wi = v, and V(G) n V(Q) = {u, v}. The labeled graph (V(G) U V(Q), E(G) U E(Q)) denote by G[u,v,Q]. Note that the edges set of G[u,v, Q] can be represented as {vo, vi,..., vn, vn+i,..., vn+m-i}, where vn+i = w2,..., v^+m-i = wm. Factually, the graph G[u,v,Q] is obtained from the graph G by "gluing" the graph Q to the vertices u and v.
Define an n-system to be a pair u = (a, ft), where a, ft : {1,...,n} a No are mappings, a(k), ft(k) < 2 + (k - 1)(m - 2) for all k = 1,..., n, and m is the number of vertices of the graph that determines the considered operation on relations (for the operation * we have m = 4 and a(k),ft(k) < 2k).
Given an n-system u = (a, ft), construct by induction the sequence of graphs Go c • • • c Gn =
= Gu. Put Go: vo- A •vi, and for k = 1,... ,n put: Gk = Gk_i[va(k), v^, G(x2kx2k+i)].
The following proposition presents the result of [11] formulated for the class R{*, c}. This result gives an infinite basis of quasi-identities for the quasi-variety Q{*, c}.
Proposition. A partially ordered semigroup (A, •, <) belongs to the quasi-variety Q{*, C} if and only if it satisfies the quasi-identity
n
(/\ pk < x2kx2k+i) ^ xi < po (14)
k=i
for every n-system u = (a, ft) and arbitrary terms po,... ,pn such that 'v1 -< G(po) and G^n'v«k) x G(pk).
Step 4. We are ready to prove the sufficiency of the conditions of the Theorem. Let u = (a, ft) be the n-system and po, pi,... ,pn be the terms such that 'v1 ^ G(po) and G^ 'vp ( k) ^ G(pk) for k = 1,.. .n. This system corresponds to the sequence graphs Go c Gi c • • • C Gn = Gw, where Gk = (Vk, Ek) for k = 0,..., n. According to the construction, for any k < n we have that
Vk = {vo, vi,..., v2k, v2k+i} and
Ek = {(vo, 1, vi)} U {(va(i), 2i, v2i+i), (v2i, 2i + 1, v^(i)) : i = 1,..., k}.
Let us Proof by induction on k that a(k) is even, P(k) is odd, and deg v2i = 0, deg+v2i > 0, deg-v2i+1 > 0, deg+v2i+1 = 0 for any i = 0,..., k.
Let k = 1. Since G0a(1)'^(1) ^ G(p1), we have that p1 = x1 or p1 = x?, and a(1) = 0, P(1) = 1, deg-v0 = 0, deg+v0 = 2, deg-v1 = 2, deg+v1 = 0, deg-v2 = 0, deg+v2 = 1, deg-v3 = 1, deg+v3 = 0.
Suppose now that it holds for k — 1, and let us show that this is true for k. Since G^' p(k) -< G(pk), according to the definition of a graph homomorphism, we get that deg+va(k) > 0, deg-v^(k) > 0. Then according to the induction assumption we get a(k) is even, P(k) is odd, and deg-v2i = 0, deg+v2i > 0, deg-v2i+1 > 0, deg+v2i+1 = 0 for any i = 0, ...,k.
Let (A, •, <) be a partially ordered semigroup satisfying identities (1) and (2). Suppose that the premise of quasi-identity (14) holds for some values of the variables x1 = a1, x2 = a2,
x3 = 0,3, ...,x2n = &2n, x2n+1 = «2n+1, i.e., Pk(a) < «2k«2k+1 for all k = 1,...,n, where a = (a1, a2 ,...,a2n+1). Let p0 = xj1 xj2 ...xjm-1 xjm. Note that Gk°'v1 -< G(p0) if and only if {xj1,xj2,...,xjm_1 ,xjm} c {x1 ,x2,...,x2k,x2k+1}, x^ = x2j+1 for some j < k such that P(j) = 1, and xj1 = x1 or xj1 = x2i for some i < k such that a(i) = 0. It follows that the equality xj1 = xjm is possible only if xj1 = xjm = x1, otherwise we can assume that all variables of p0 are different.
Let max(p0) be the greatest k such that at least one of the variables x2k or x2k+1 is included in the term p0. Let us prove by induction on max(p0) that a1 < p0(a). If max(p0) = 0 then p = x1 or p0 = x2. Thus, using identity (1) we obtain a1 < p0(a). Suppose now that a1 < p0(a) holds for max(p0) = k — 1, and let us show that this is true for max(p0) = k.
If both variables x2k,x2k+1 are included in p0, then the following cases are possible:
1) p0 = x2kx2k+1 xj3... xjm-1 xjm, then using the induction assumption we get
a0 ^ Pk (0)aj3 . . . °jm _ 1 j < a2k a2k+1 j . . . ajm-1 j = P0(0);
2) p0 = xj1 x2kx2k+1 xj4.. .xjm-1 xjm, then using the induction assumption we get
a0 < j Pk (0)aj4 . . . °jm_1 j < j a2k a2k+1 0j4 . . . °jro_1 °jm = P0(0);
3) p0 = xj1 xj2.. .xjm-3x2kx2k+1, then using the induction assumption we get
°0 < j j . . . ajm_3Pk (0) < j j . . . °jm_3 °2k°2k+1 = P0 (0);
4) p0 = x2kxj2 ... xjm_1 x2k+1, then a(k) = 0 and P(k) = 1. It follows that
Gk° 'V1 ^ G(Pk xj2 . . . x jm _ 1 Pk ), and by the induction assumption we get
(2)
00 < Pk (a)aj2 ... a jm _ 1 Pk (a) < a2k a2k+1 aj2 ... ajm_1 a2k a2k+1 < < a2kaj2 ... ajm_1 a2k+1 = P0(a).
If only one of the variables x2k or x2k+1 is included in p0, then the following cases are possible:
5) p0 = x2kxj2... xjm_1 xjm, then using the induction assumption we get
(2)
a0 < Pk (a)aj3 . . . a jm _ 1 °jm < °2k a2k+1 j . . . ajm _ 1 j ^ °2k j . . . °jm_1 j = P0 (a);
6) p0 = xj x2kxj3.. .xjm_1 xjm, then using the induction assumption we get
(2)
a0 < j Pk (a)aj3 . . . a jm _ 1 °jm < j °2k a2k+1 j . . . ajm _ 1 j ^ j °2k j . . . °jm_1 j = P0 (a);
7) po = xj x2k+ixj3.. .xjm-1 xjm, then using the induction assumption we get
(2)
ao < j pk (a)aj4 . . . ajm-1 j < j a2ka2k+i j . . . ftjm-1 ftjm <
< j a2k+i j . . . ajm-1 j = po(a);
8) po = xj xj2.. .xjm-1 x2k+i, then using the induction assumption we get
(2)
ao < flji aj2 ... ajm-1 pk (a) < a^ a^ ... ajm-1 a2k «2k+i < a^ a^ ... ajm-1 a2k+i = po (a).
Thus, we have proved that the partially ordered semigroup (A, •, <) satisfies quasi-identities (14). Therefore, according to the Proposition we have (A, •, <) G Q{*, c}. This completes the proof of the Theorem.
Step 4. Let us prove the sufficiency of the conditions of Corollary 1. Suppose that a semigroup (A, •) satisfies identities (3)-(5) and A2 = {a2 : a G A}. We define the relation < on the set A by setting
^ = {(x, y) G A x A2 : x2 = yxy} U {(x, x) G A x A : x G A}.
Let us show that (A, •, <) is the partially ordered semigroup satisfying identities (1) and (2). The reflexivity of the relation < follows directly from the definition.
To prove the transitivity assume that x < y and y < z. Without loss of generality, we can suppose that x = y and y = z. Then x2 = yxy, y2 = zyz and y2 = y, z2 = z, hence
2 i11) 2 (12) . . T, .. , ...
x2 = yxy = zyzxzyz = zzyxyzz = zyxyz = zx2z = zxz, i.e., x < z. Thus, < is transitive.
Assume that x < y, y < x and x = y. Then x2 = yxy, y2 = xyx and x2 = x, y2 = y, hence
x = x2 = yxy = y2xy2 = xyxxxyx (= xyx = y2 = y. This contradicts the assumption x = y. Thus, < is a partially order relation.
Let us show that the relation < is compatible with multiplication. Suppose that x < y and
, T, 2 i 2 i f x2 (3) (13) 2 (12) 3 (11)
x = y. I hen x2 = yxy and y2 = y, hence (xz)2 = xz = x2z = yxyz = yxyz3 = yzxzyz
and (yz)2 == yz. Thus, xz < yz. Further, (zx)2 = zx (=) zx2 = zyxy (=) z3yxy (=) zyzxzy and
(zy)2 == zy. Thus, zx < zy.
Since x2 (=) x2xx2 and (x2)2 = x2, we have x < x2. Since (xyz)2 = xyz (=) xyz3 (=)
= x3yz3 (=) xzxyzxz and (xz)2 == xz , we have xyz < xz. Therefore, (A, •, <) satisfies identities (1) and (2), hence (A, •, <) G Q{*, c} and (A, •) G Q{*}. This completes the proof of Corollary 1. Step 5. Let us prove the sufficiency of the conditions of Corollary 2 and 3.
Lemma 3. Let {Uj : j G J} be a family of pairwise non-intersecting sets and U = = U{Uj : j G J}. If a partially ordered semigroup (A, •, <) is a subdirect product of a family {($j, *, c) : j G J} of partially ordered semigroups of relations on Uj, and satisfies identity (6), then (A, •, <) isomorphically embedded in (Rel(U), *, c).
Proof. Let ^j : A —> be the corresponding surjective homomorphisms from A on the components of the direct product f]{$j : j G J}. According to the properties of homomorphic images, we see that all components , *, c) satisfy identity (6). Hence, for all j G J we have 0 G ^j or = {0}. It follows that (A, •, <) is subdirect product of the family {($j, *, c) : j G Jo}, where Jo = {j G J : 0 G }.
For given a G A, put = (a). We define a mapping ^ : A —Rel(U) in the following way. Put ^(a) = U{pri: j G Jo}xU{pr2: j G Jo}, if a2 = a, and ^(a) = U{pa : j G Jo}UUMb) : b2 = b < a} otherwise. Let us show that ^ is an isomorphic embedding (A, •, <) in (Rel(U), *, c).
Note that ^(a) n Uj x Uj = for all a G A. It follows that ^(a) c ^(a) if and only if a < b. Forver, since (ab)2 = ab, we have
<^(ab) = У{pripf : j G Jo} x у {pr2pf : j G Jo}
= U{Pr1 (Pr1Pa x Pr2Pj) : j G J0} x U{Pr2 (Pr1 p? x Pr2pb) : j G J0} = = (U{Pr1 P? : j G J0} x y{Pr2pb : j G J0}) = pn<p(a) x Pr2<p(6) = <p(a) * <p(6).
□
Lemma 4. Suppose that (A, •, <) satisfies identities (1) and (6). Then (A, •, <) belongs to
R{*, c}.
Proof. If (A, •, <) satisfies identities (1) and (6), then according to the Theorem we have (A, •, <) G Q{*, c}. In respect that the class R{*, c} is axiomatizable [34], we obtain that (A, •, <) is a subdirect product of a family of partially ordered semigroups from R{*, c}. Hence, according to Lemma 3, we obtain that (A, •, <) belongs to R{*, c}. □
Lemma 5. Suppose that (A, •) satisfies identity (6). Then (A, •) belongs to R{*}.
Proof. If (A, •) satisfies identity (6), then according to Lemma 1 it also satisfies identities (3)-(5). Let < be the partial order relation constructed in the proof of Corollary 1. Then by Lemma 4, we have (A, •, <) G R{*, c}. Therefore, (A, •) G R{*}. □
Lemma 6. Suppose that (A, •) contains the zero element o and satisfies axiom (7). Then (A, •) satisfies identities (6) or a6 = o for all a, 6 = o.
Proof. If there exist elements a = o and 6 = o such that a6 = o, then for all x, y = o we
have xay == xy, xby == xy, and xaby = o, hence xy == xyxy = xayxby == xaby = xoy = o. It follows that xyz = xz for all x, y, z G A, i.e., (A, •) satisfies identities (6). □
Suppose that (A, •, <) contains the zero element o and satisfies identity (1) and axioms (7) and (8). Put B = A\{o}. According to Lemmas 4 and 6, we can suppose that xy G B for all x, y G B, and (B, •, <) satisfies identities (1) and (6), hence (B, •, <) belongs to R{*, c}. It means that there exists an isomorphism F from the partially ordered semigroup (B, •, <) to some partially ordered semigroup of relations ($, *, c) and 0 G Putting F(o) = 0, we get the isomorphism from (A, •, <) to U {0}, *, c). Therefore, (A, •, <) belongs to R{*, c}. This completes the proof of Corollary 2.
Suppose that (A, •) contains the zero element o and satisfies axiom (7), B = A\{o}, and let < be the partial order relation on B constructed in the proof of Corollary 1. Extend the relation < on A by putting o < a for all a G A. Then (A, •, <) satisfies the condition 3 of the Theorem, hence (A, •, <) G R{*, c}. Therefore, (A, •) belongs to R{*}. This completes the proof of Corollary 3.
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Поступила в редакцию / Received 09.03.2023 Принята к публикации / Accepted 26.04.2023 Опубликована / Published 30.08.2024
