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2021 Теоретические основы прикладной дискретной математики №53
ТЕОРЕТИЧЕСКИЕ ОСНОВЫ ПРИКЛАДНОЙ ДИСКРЕТНОЙ МАТЕМАТИКИ
UDC 512.53 DOI 10.17223/20710410/53/1
EQUATIONS OVER DIRECT POWERS OF ALGEBRAIC STRUCTURES
IN RELATIONAL LANGUAGES1
A. Shevlyakov
Sobolev Institute of Mathematics SB RAS, Omsk, Russian Federation Omsk State Technical University, Omsk, Russian Federation
E-mail: art.shevlyakov@gmail.com
For a semigroup S (group G) we study relational equations and describe all semigroups S with equationally Noetherian direct powers. It follows that any group G has equationally Noetherian direct powers if we consider G as an algebraic structure of a certain relational language. Further we specify the results as follows: if a direct power of a finite semigroup S is equationally Noetherian, then the minimal ideal Ker(S) of S is a rectangular band of groups and Ker(S) coincides with the set of all reducible elements.
Keywords: relations, groups, semigroups, direct powers, equationally Noetherian algebraic structures.
Introduction
Let A be an algebraic structure of a functional language L with a universe A. In other words, there are certain functions and constants over A that correspond to symbols of L. One can define a structure Pr(A) with the universe A of a pure relational language Lpred as follows:
Rf (xi, ...,xn ,y) = {(xi, ...,xn,y): f (xi, ...,xn) = y G A}; Rc(x) = {x : x = c G A},
where functional and constant symbols f, c belong to the language L. Namely, the relation Rf G Lpred (Rc G Lpred) is the graph of a function f (respectively, constant c).
The Lpred-structure Pr(A) is called the predicatization of an L-structure A. In particular, if A is a group of the language Lg = {-,-i, 1}, then Pr(A) is an algebraic structure of the language Lg_pred with the following relations:
M(x,y,z) ^ xy = z; (1)
I(x,y) ^ x = y_1; (2)
E(x) ^ x = 1. (3)
1The author was supported by the RSF-grant 18-71-10028 (Theorem 1) and RSF-grant 19-11-00209 (Theorem 4).
Notice that any equation over a group A may be rewritten in the language Lg_pred by the introducing new variables. For example, the equation x_1y_1xy = 1 has the following correspondence in the relational language Lg_pred:
Pr(S)
I(x, X\),
1 (У,Уl), M (xi,yi,zi), M(zi,x, z2), M (Z2 ,y,za), IE (Z3).
It is easy to see that the projection of the solution set of S onto the variables x,y gives the solution set of the initial equation x-iy-ixy = 1. More generally, for any finite set of group equations S in variables X there exists a system Pr(S) of equations in the language Lg_pred such that the solution set of S is the projection of the solution set Pr(S) onto the variables X. Hence, there arises the following important problem.
Problem. What properties of a finite system S are determined by the system Pr(S)?
This problem was originally studied in [1], where it was proved the general results for relational structures Pr(S).
We study equations over direct products of semigroups. Namely, for a finite semigroup S we give necessary and sufficient condition whether the direct power n Pr(S) is equationally Noetherian. It continues the research [2], where we found the necessary and sufficient conditions for the equationally Noetherian property of direct powers of functional algebraic structures (groups, rings, monoids). For example, a group (ring) has equationally Noetherian direct powers in a functional language with constants iff it is abelian (respectively, with zero multiplication).
On the other hand, we prove below that any finite group in the language Lg-pred has equationally Noetherian direct powers (Corollary 1). Moreover, the similar result holds for the natural generalizations of groups: quasi-groups and loops (Remark 1).
However, the class of semigroups has a nontrivial classification in the relational language. We find two quasi-identities
VaVbVaV/ ((aa = a/) ^ (ba = b/)); (4)
VaVbVaV/ ((aa = /a) ^ (ab = /b)) (5)
such that a finite semigroup S satisfies (4), (5) iff any direct power of Pr(S) is equationally Noetherian (Theorem 1).
In the class of finite semigroups the conditions (4), (5) imply that the minimal ideal (kernel) of a semigroup S is a rectangular band of groups, and the kernel Ker(S) (the minimal ideal of S) coincides with the ideal of reducible elements of S. If the kernel of a finite semigroup S is a group, then the converse statement also holds (Theorem 4). However, the converse statement is not true in general (Example 1).
1. Basic notions
An algebraic structure of the language Ls-pred = {M(3)} (Lg-pred = {M(3),I(2),E(i)}) is called the predicatization of a semigroup S (group G) if the operations over S (G) corresponds to the relations (1)-(3). The predicatization of a semigroup S (group G) is denoted by Pr(S) (respectively, Pr(G)).
Following [3], we give the main definitions of algebraic geometry over algebraic structures (below L G {Ls-pred, Lp-pred})-
An equation over L (L-equation) is an atomic formula over L. The examples of equations are the following: M(x,x,x), M(x,y,x) (Ls-pred-equations); M(x,x,y), I(x,y), I(x,x), E(x) (Lg-pred-equations).
A system of L-equations (L-system for shortness) is an arbitrary set of L-equations. Notice that we will consider only systems in a finite set of variables X = {xi,x2,... , xn}. The set of all solutions of S in an L-structure A is denoted by Va(S) C An. A set Y C An is said to be an algebraic set over A if there exists an L-system S with Y = Va(S). If the solution set of an L-system S is empty, S is said to be inconsistent- Two L-systems S1, S2 are called equivalent over an L-structure A if Va(S1) = Va(S2).
An L-structure A is L-equationally Noetherian if any infinite L-system S is equivalent over A to a finite subsystem S' C S.
Let A be an L-structure. By L(A) we denote the language L U {a : a G A} extended by new constants symbols which correspond to elements of A. The language extension allows us to use constants in equations. The examples of equations in the extended languages are the following: M(x,y,a) (Ls-pred(S)-equation and a G S); M(a,x,b), I(x,a), E(a) (Lg-pred(S)-equations and a,b G G). Obviously, the class of L(A)-equations is wider than the class of L-equations, so an L-equationally Noetherian algebraic structure A may lose this property in the language L(A).
One can directly prove that any finite L(A)-structure A is always L(A)-equationally Noetherian.
Since the algebraic structures A and Pr(A) have the same universe, we will write below
VA(S) (L(A)) instead of VPr(A)(S) (respectively, L(Pr(A))).
Let A be a relational L-structure. The direct power nA = n A of A is the set of all
iei
sequences [ai : i G I] and any relation R G L is defined as follows
R([a(1) : i G I], [a(2) : i G I],..., [a(n) : i G I]) ^ R(a((1), a(2),... , a(n)) for each i G I.
A map nk: nA "A is called the projection onto the i-th coordinate if nk([ai : i G I]) = ak.
Let E(X) be an L(nA)-equation over a direct power nA. We may rewrite E(X) in the form E(X, C), where (C is an array of constants occurring in the equation E(X). One can introduce the projection of an equation onto the i-th coordinate as follows:
Ki(E(X)) = ni(E(X, "(C)) = E(X, ffi((C)),
where ni((C) is an array of the i-th coordinates of the elements from ". For example, the Ls-pred(nA)-equation M(x, [a1, a2, a3,...], [b1, b2, b3,...]) has the following projections
M(x, a1, b1), M(x, a2, b2), M (x,a3,b3),
Obviously, any projection of an L(nA)-equation is an L(A)-equation.
Let us take an L(nA)-system S = {Ej (X) : j G J}. The i-th projection of S is the L(A)-system defined by ni(S) = {ni(Ej(X)) : j G J}. The projections of an L(nA)-system S allow to describe the solution set of S by
VnA(S) = {[Pi : i G I] : Pi G VaMS))}.
(6)
In particular, if one of the projections n^(S) is inconsistent, so is S.
The following statement immediately follows from the description (6) of the solution set over a direct powers.
Lemma 1. Let S = {Ej (X) : j G J} be an L(nA)-system over nA. If one of the projections nj(S) is inconsistent, so is S. Moreover, if A is L(A)-equationally Noetherian, then an inconsistent L(nA)-system S is equivalent to a finite subsystem.
Proof. The first assertion directly follows from (6). Suppose A is L-equationally Noetherian, and n^(S) is inconsistent. Hence, n^(S) is equivalent to its finite inconsistent subsystem {^(Ej(X)) : j G J'}, | J'| < ro, and the finite subsystem S' = {Ej(X) : j G G J'} C S is also inconsistent. ■
2. Predicatization of semigroups and groups
Theorem 1. Let Pr(S) be the predicatization of a finite semigroup S. A direct power of Pr(S) is Ls-pred(nS)-equationally Noetherian iff the quasi-identities (4), (5) hold in S.
Proof. First, we prove the "if" part of the theorem. Suppose S satisfies (4,5) and consider an infinite Ls-pred(nS)-system S. One can represent S as a finite union of the following systems
S = U Scij U SiCj U Sijc U Scci U Scic U SiccU SCb (7)
i^ij^ra i^i^ra i^i^ra i^i^ra
where each equation of So is one of the following types:
1) xi xj;
2) xi cj;
3) ci - cj;
4) M (xi, xj ,xfc);
5) M(ci, Cj, cfc),
and SCij = {M(cfc, xi, xj) : k G K}, SiCj = {M(xi, cfc,xj) : k G K}, SjC = {M(xi,xj, cfc) : k G K}, SCCi = {M(cfc, dfc,xi) : k G K}, SCiC = {M(cfc,xi, dfc) : k G K}, SiCC = = {M(xi, ck, dk) : k G K} (ck, dk G nPr(S)), where each system above has its own index set K.
Clearly, the system So is equivalent to its finite subsystem. So it is sufficient to prove that the other systems are equivalent to their finite subsystems. According to Lemma 1, we may assume that all systems below are consistent.
Thus, we have the following cases:
1) Let Sicc = {M(xi, ck, dk) : k G K} and M(xi, ci, di) be an arbitrary equation of Sicc. Since Sicc is consistent, then one can choose a G Vns(Sicc), 3 G VnSs(M(xi, ci, di)). We have <5ci = /^ci = di. By the quasi-identities (4), (5), ack = for any ck. Hence, 3 satisfies all equations from Sicc, Thus, Sicc is equivalent to the equation M(xi, ci, di). The proof for the system Scic is similar.
2) Let Scci = {M(ck, dk,xi) : k G K}. Since the system Scci is consistent, the products ckdk are equal to each other, hence c = ckdk for all k G K. Thus, the whole system Scci is equivalent to any equation M(ck, dk,xi).
3) Let Sicj = {M(xi, ck,xj) : k G K} (the proof for Scij is similar). Since Sicj is consistent, there exist a point (a, 3) G Vns(Sicj) and the equalities ack = aici = / hold for any k,/ G K. By (4), (5), for any 7 G nS it holds 7ck = Yycj. Thus, the solution set of Sicj is Y = {(7,7cko) | 7 G nS} for a fixed k0 G K. Thus, Sicj is equivalent to the equation M(xi, cko ,xj).
4) Let Sijc = {M(xi,xj, ck) : k G K}. Since the system Sijc is consistent, the elements ck (k G K) are equal to each other. Hence, the system Sijc consists of the same equations. Thus, Sijc is equivalent to any equation M(xi,xj-, ck). Now, we prove the "only if" part of the theorem. Suppose the quasi-identity (4) does not hold in S (for the formula (5) the proof is similar). It follows there exist elements a,b,a,ft
satisfies the first n equations of S (since we obtain the true equalities aft = c or bft = bft). However the (n + 1)-th equation of S gives an+1a = cn+1, since its (n + 1)-th projection defines the equation bx = bft, but ba = bft. Thus, S is not equivalent to any finite subsystem. ■
Corollary 1. Let Pr(G) be the predicatization of a finite group G. Then any direct power of Pr(G) is Lg-pred(nG)-equationally Noetherian.
Proof. Since the equality aa = aft (aa = ft a) implies a = ft in any group, the quasi-identities (4), (5) obviously hold in G. Thus, any infinite system of the form {M(*, *, *) : i G I} is equivalent to a finite subsystem.
One can directly prove that for any finite group G the infinite systems of the form {I(*, *) : i G I} ({E(*) : i G I}) are also equivalent to their finite subsystems over nG.
Thus, any system of Lg-pred(nG)-equations is equivalent over nG to its finite subsystem. ■
Remark 1. The Corollary 1 also holds for finite quasi-groups. Notice that a quasi-group is a non-associative generalization of a group. Any quasi-group admits the analogue of divisibility, hence the quasi-identities (4), (5) obviously hold in any quasi-group. Thus, any direct power of a quasi-group G is Ls-pred(nG)-equationally Noetherian (here we consider quasi-groups and loops in the language Ls-pred, since not any quasi-group admits the relations I(x,y) and E(x)).
Below we study finite semigroups S that satisfy Theorem 1.
A subset I C S is called a left (right) ideal if for any s G S, a G I it holds sa G I (as G I). An ideal which is right and left simultaneously is said to be two-sided (or an ideal for shortness).
A semigroup S with a unique ideal I = S is called simple. Let us remind the classical Sushkevich — Rees theorem for finite simple semigroups.
Theorem 2. For any finite simple semigroup S there exist a finite group G and finite sets I, A such that S is isomorphic to the set of triples (\,g,i), g G G, A G A, i G I. The multiplication over the triples (A,g,i) is defined by
such that aa = aß = c, ba = bß. Let us consider the system
S = {M(an,x, cn) : n G N},
n times
(X,g,i)(ß,h,j) = (X,gpißh,j),
where piß G G is an element of a matrix P such that
1) P consists of |11 rows and |A| columns;
2) the elements of the first row and the first column equal 1 G G (i.e., P is normalized)).
Following Theorem 2, we denote any finite simple semigroup S by S = (G, P, A, I).
The minimal ideal of a semigroup S is called a kernel and denoted by Ker(S) (any finite
semigroup always has a unique kernel, and the kernel is always simple, i.e., Ker(S) satisfies Theorem 2). Obviously, if S = Ker(S), then the semigroup is simple. If Ker(S) is a group, then S is said to be a homogroup. The next theorem contains the necessary information about homogroups.
Theorem 3 [4]. In a homogroup S the identity element e of the kernel Ker(S) is idempotent (e2 = e) and belongs to the center of S (i.e., e commutes with any s G S).
A semigroup S is called a rectangular band of groups if S = (G, P, A, I) and piA = 1 for any i G I, A G A.
Lemma 2. Suppose a finite simple semigroup S satisfies (4), (5). Then S is a rectangular band of groups.
Proof. By Theorem 2, S = (G, P, A,1) for some finite group G, matrix P and finite sets of indexes A, I.
Assume that |A| > 1 and piA = 1 for some i, A.
Let a = (1,1,1), a = (A, 1,1), 3 = (1,1,1) and hence
aa = (1,1,1)(A, 1,1) = (1,1,1) = (1,1,1)(1,1,1) = a/. (8)
However, for b = (1,1, i) we have
ba = (1,1, i)(A, 1,1) = (1,PiA, 1) = (1,1,1) = (1,1,i)(1,1,1) = b/. (9)
Thus, the equalities (8), (9) contradict (4), (5). ■
An element s of a semigroup S is called reducible if there exist a, b G S with s = ab. Clearly, the set of all reducible elements Red(S) is an ideal of a semigroup S.
Lemma 3. Let S be a finite semigroup satisfying (4), (5). Then Ker(S) is the set of all reducible elements.
Proof. Since the kernel Ker(S) is simple, Theorem 2 gives Ker(S) = (G, P, A, I) for some finite G, P, A,1. Let b G S. We have (A, g,i)b = (A,g,i)(1,1,i)b = (A, g,i)r, where r = (1,1,i)b G Ker(S). By (4), we obtain ab = ar for any a G S. Since ar G Ker(S), so is ab. Thus, any product of elements belongs to Ker(S), hence Red(S) = Ker(S). ■
Theorem 4. If Ker(S) = Red(S) for a finite homogroup S, then S satisfies (4), (5) or, equivalently, nS is Ls-pred(nS)-equationally Noetherian.
Proof. Let us take a,b,a,3 such that aa = a/, and e be the identity of Ker(S). We have
aa = a3 | -e, eaa = ea3,
(ea)a = (ea)3 | -(ea)-i since ea belongs to the group Ker(S), ea = e3 | e is a central element, ae = 3e.
We have (below we use ba,b3 G Ker(S) = Red(S)):
ba = (ba)e = b(ae) = b(3e) = (b3)e = b3.
Thus, the quasi-identity (4) holds for S. The proof for the quasi-identity (5) is similar. ■
One can directly check that for a rectangular band of groups S = (G, P, A,1) the analogue of Theorem 4 also holds.
Thus, there arises the following question.
Question. Suppose the kernel Ker(S) of a finite semigroup S satisfies the following conditions:
1) Ker(S) = Red(S);
2) Ker(S) is a rectangular band of groups.
Does S satisfy the quasi-identities (4), (5)?
Example 1. The answer for the last question is negative. Let us consider a semigroup S with the following multiplication table:
a b Zi Z2 Z3 Z4
a Z4 Z4 Z2 Z4 Z4 Z4
b Z4 Z4 Z3 Z4 Z4 Z4
Zi zi Zi Zi Zi Zi Zi
Z2 Z2 Z2 Z2 Z2 Z2 Z2
Z3 Z3 Z3 Z3 Z3 Z3 Z3
Z4 Z4 Z4 Z4 Z4 Z4 Z4
This Table defines an associative binary operation (we checked it by a computer). One can directly compute that Ker(S) = Red(S) = (zi, z2, z3, z4}. Since the elements z are left zeros, we have Ker(S) = (G, P, A,/), where G = (1}, P = (1,1,1,1), A = = (1, 2, 3, 4}, / = (1}. However, the quasi-identity (5) does not hold in S, since az1 = bz1, but az0 = bz0.
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