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D. A. Bredikhin
ON SEMIGROUPS OF RELATIONS WITH THE OPERATION OF REFLEXIVE-RESTRICTIVE
MULTIPLICATION
In this paper, we present the proof of announced in [1] results concerning the class of semigroups of relations with the operation of reflexive-restrictive multiplication. Some other results concerning classes of semigroups and groupoids of relations with Diophantine operations [2,3] can be found in
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. Let Rel(U) be the set of all binary relation on U. We concentrate our attention on the associative operation * on Rel(U) that is defined as follows: for any relations p and Œ from Rel(U), put
Note that the result of this operation coincides with the restriction of the second multiplier on the reflex projection of the first one. For this reason, we will treat this operation as operation of reflexive-restrictive multiplication. It is clear that the set-theoretic inclusion C is compatible with respect this operation. Denote by R{*} (respectively, C}) the class of all semigroups (partially ordered semigroups) isomorphic to semigroups (partially ordered semigroups) of relations with the operation of reflexive-restrictive multiplication.
Theorem. The class R{*, c} forms a quasi-variety and does not form a variety in the class of all partial ordered semigrops. A partially ordered semigroup (A, •, <) belongs to the class R{*, c} if and only if it satisfies the folloing identities and quasi-identities:
Received 7 September 2020
[1, 4-12].
p * Œ = {(x, y) G U x U : (x,x) G p A (x,y) G œ}.
x2y = xy, xyz = yxz,
xy < y,
x < yz ^ x < yx.
(1) (2)
(3)
(4)
Corollary 1. A partially ordered semigroup (A, •, <) belongs to the variety generated by the class R{*, c} if and only if it satisfies identities
(1) - (3).
Corollary 2. The class R{*} forms a variety. A semigroup (A, •) belongs to R{*} if and only if it satisfies identities (1) - (2).
Proofs. Necessity of conditions (1) - (4) is carried out by direct verification. To prove sufficiency, assume that (A, •, <) satisfies the conditions (1) - (4). For given xG A, put Ux = {(x, 0), (x, 1)}. Let us define the mapping Fx : A ^ Re/(Ux) in the following way:
1) ((x, 0), (x, 1)) G Fx(a) if and only if x < a;
2) ((x, 0), (x, 0)) G Fx(a) if and only if x < ax.
Put U = |J{Ux : x G A} and F(a) = |J{Fx(a) : x G A}. It is clear that a < b implies F(a) C F(b). Suppose that F(a) C F(b). Then Fa(a) C Fa(b). Since ((a, 0), (a, 1)) GFa(a), we have ((a, 0), (a, 1)) GFa(b), hence a<b. Thus, a < b if and only if F(a) C F(b).
Let us show that F(ab)CF(a)*F(b). Suppose that ((x, 0), (x, 1))gF(ab), then ((x, 0), (x, 1)) G Fx(ab). Hence x < ab. Using (4) we obtain x < ax, and using (3) we have x<b. Hence ((x, 0), (x, 0))GF(a) and ((x, 0), (x, 1))GF(b), i.e., ((x, 0), (x, 1)) GF(a)*F(b). Further, suppose that ((x, 0), (x, 0))GF(ab), then ((x, 0), (x, 0)) G Fx(ab). Hence x < abx. Using (3) we obtain x < ax and x < bx. It follows that ((x, 0), (x, 0)) G F(a) and ((x, 0), (x, 0)) G F(b), i.e., ((x, 0), (x, 0)) G F(a) * F(b). Conversely, assume that ((x, 0), (x, 1)) G G F(a) * F(b), then ((x, 0), (x, 1)) G Fx(a) * * Fx(b). Hence x <ax and x < b. It follows that x <ax < ab, i.e., ((x, 0), (0,1)) G F(ab). Further, suppose that ((x, 0), (x, 0)) G F(a) *F(b), then ((x, 0), (x, 0)) G Fx(a) *Fx(b). Hence x<ax and x < bx. It follows that x < ax < abx, i.e., ((x, 0), (x, 0)) G F(ab). Thus, F is an isomorphism of (A, •, <) into (Re/(U), *, C), i.e., (A, •, <) belongs to
R{*, c}.
Show that the classes class R{*, c} does not form a variety in the class of all partial ordered semigrops. Let us consider the groupoid on the set A = {a, b, c, d} given by the following table
a b c d
a b b c d
b b b c d
c c c d d
d c c d d
By a direct check, we verify that this groupoid forms a semigroup and satisfies identities (1) and (2). We set on A the partial order relation < by putting d < c < b < a. By a direct check, we make sure that this relation
is compatible with multiplication and identities (3) is satisfied. On the other hand we have b < dd and db = c < b, i.e., quasi-identity (3) is not satisfied. Therefor, the class R{*, c} does not form a variety in the class of all partial ordered semigroups. This completes the proof of the theorem.
To prove Corollary 1, note that it is not difficult to show by direct verification that a free partially ordered semigroup of the variety defined by identities (1) - (3) satisfies quasi-identity (4). It follows that the identities (1) - (3) form a basis of the variety generated by the class R{*, c}.
To prove Corollary 2, let us suppose that (A, •) satisfies the identities (1) and (2). Define on A the relation <: a < b if and only if a = b or a = cb for some c G A. It is easy to check that (A, •, <) is a partially ordered semigroup satisfying the conditions (3) and (4). It folows that (A, •) belongs to R{*}.
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