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5
D. A. Bredikhin
ON SEMIGROUPS OF RELATIONS WITH PRIMITIVE-POSITIVE OPERATIONS OF RANK TWO
Received 10 May 2019
Let Rel(U) be the set of all binary relations on a base set U. A set of relations $ C Rel(U) closed with respect to some collection Q of operations on relations forms an algebra Q) called an algebra of relations. Theory of algebras of relations is an essential part of modern algebraic logic and has important applications in theory of semigroups [1].
Operations on relations are usually determined using first-order predicate calculus formulas. Such operations are called logical. For any formula p(z0, z1,r1,r2) of the first-order predicate calculus (without equality) with the set of free variables included in {z0, z1} and having two binary predicate symbols r1, r2, we can associate the binary operation F on Rel(U) defined in the following way: F(p1, p2) = {(u, v) E U x U : p(u, v, p1? p2)}, where p(u, v, p1? p2) means that the formula p holds whenever z0, z1 are interpreted as u, v, and r1?r2 are interpreted as relations p1? p2 E Rel(U).
An operation on relations is called primitive-positive [2] (in other terminology - Diophantine [3]) if it can be defined by a formula of the firstorder predicate calculus containing in its prenex normal form only existential quantifiers and conjunctions. We say the operation has rank k if it can be defined by a formula containing k conjunctive members and cannot be determined by formulas with a smaller number of them. Note that the set-theoretical inclusion C is compatible with all primitive-positive operations. Thus, any algebra of relations Q) with primitive-positive operations can be considered as partially ordered Q, C).
The operation F*(p1? p2) = F(p2, p1) = {(u, v) E UxU : p(u, v, p2, p1)} is called dual to the operation F. The abstract properties of these operations are dual to each other. For this reason, we will consider only one of these operations. Let p be the formula obtained from the formula p by replacing in it atomic formulas rk(zi5 Zj) by formulas rk(zj, z^). The operation F(pb p2) = {(v,u) E U x U : p(u,v, p2, p1)} is called conjugate to the operation F. The operation is called self-conjugate if F = F*. Note that the mapping f (p) = p-1 is an antiisomorphism of partially ordered algebras (Rel(X), F, C) and (Rel(X), F, C), hence it will be enough to limit consideration to only one of these operations.
Denote by R{^} (respectively, R{^, C}) the class of all algebras (partially ordered algebras) isomorphic to the ones whose elements are binary relations and whose operations are members of Let V(VC}) be the variety and let Q{^} (Q{^, C}) be the quasi-variety generated by R{^} (generated by R{^, C} in the class of all partially ordered algebras of the corresponding type). The following problems naturally arise when the class R{ft} (R{fi, c}) is considered.
1. Find a system of axioms for the class R{^} (R{^, C}).
2. Find a basis of quasi-identities for the quasi-variety Q{^} (Q{^, C}).
3. Find a basis of identities for the variety V(VC}).
4.0 Does the class R{^} (R{^, C}) form a quasi-variety?
5. Does the quasi-variety Q{^} (Q{^, C}) form a variety?
This paper presents the results of solving problems 1-5 for all classes of algebras and partially ordered algebras of relations with one binary associative primitive-positive operation of rank 2, i.e, for classes of semigroups of relations. Their proofs are based on the description of quasi-equational theories of algebras of relations with primitive-positive operations [3].
The operations of intersection R and relational product o are primitive-positive operations of rank 2. It is well known that the class R{R} coincides with the class of all semilattices and the class R{o} coincides with the class of all semigroup. We exclude these operations from our further consideration.
Now, we give a list of all binary primitive-positive operations of rank 2 (except for R, o, and operations dual and conjugate to the ones listed below). This list can be obtained by means of a rune check for the associativity of all binary primitive-positive operations of rank 2.
F1(p1, P2) = v)
F2(P1, P2) = v)
F3(P1, P2) = v)
F4(P1, P2) = v)
F5(P1, P2) = v)
Fo(P1, P2) = v)
Fr(p1, P2) = v)
Fg(p1, P2) = {(u> v)
Fg(p1, P2) = v)
F1o(p1, P2) = = {(u ,v
G U x U : (3s)(u, s) G Pi A (u,v) G p2}, : (u,u) G pi A (u,v) G P2}, G U x U : (3s, t)(u, s) G p1 A (u, t) G p2}, G U x U : (u,u) G p1 A (u,u) G p2}, G U x U : (3s,t)(s,t) G p1 A (u,v) G p2}, : (3s,t, w)(s,t) G pi A (w,v,) G P2}, G U x U : (3/, s,t, w)(/, s) G p1 A (t, w) G p2}, G U x U : (3s)(s, s) G p1 A (v,v) G p2}, G U x U : (3s,t)(s,s) G p1 A (t,t) G p2}, = {(u,v) G U x U : (3s,t)(u, s) G p1 A (t,v) G p2}.
Note that operations F5,F7,F10 are self-conjugate. Put Rk = R{Fk}, Rf =
= R{Fk, C}, Qk = Q{Fk}, Qf = Q{Fk, C}, Vk = V{Fk}, Vs = V{Fk, C}
<
for 1 < k < 10. Let us consider the following identities and axioms:
x2 = x (1), x2y = xy (2), xy2 = xy (3), xy = yx (4), xyz = yxz (5), xyz = xyxz (6), xyzx = xzyx (7), xy < y (8), xy < y2 (9), x < x2 (10), xyz < xz (11), xy = y V xz = zx = x (12), xy = y2 V xz = zx = x(13), xy = y2 V xz = zx = x2(14), xyz = xz V yt = ty = y (15), xy = yx = x ^ x < z (16), xy = yx = x ^ x2 < z (17).
Denote by ModjS} the class of semigroups (respectively, partially ordered semigroups) satisfying the system of axioms £. The following results solve problems 1-5 for classes Rk and R<.
1- Qk = Vk and Q< = Vk< for all k; Rk = Qk and R< = Q< for k = 1,..., 4.
2. Vi = V5 = Modk{(1),(5)} and V< = V5< = Mod{(1),(5), (8)}.
3. V2 = Mod{(2), (5)} and V2< = Mod{(2), (5), (8)}.
4. V3 = V4 = V7 = V9 = Mod{(2), (4)} and Vo = Vg = Mod{(2), (3), (5)}.
5. V3< = V7< = Mod{(2), (4), (9), (10)} and V4< = V9< = Mod{(2), (4), (9)}.
6. V6< = M^od{(2), (3), (5), (9), (10)} and Vg< = Mod{(2), (3), (5), (9)}.
7. V10 = Mod{(2), (3), (6), (7)} and = Mod{(10), (11)}.
8. R5 = Mod{(12)} and R< = Mod{(12), (15)}.
9. Ro = Mod{(13)} and R< = Mod{(10), (13), (16)}.
10. R7 = Mod{(4), (13)} and R< = Mod{(4), (10), (13), (16)}.
11. Rg = Mod{(13)} and R< = Mod{(13), (17)}.
12. R9 = Mod{(4), (13)} and R9< = Mod{(4), (13), (17)}.
13. R10 = Mod{(15)} and R<0 = Mod{(10), (15), (16)}.
The class R1 has been reviewed and characterized in [4]. Characteristics of the classes R2, R3, R4, R10 was announced in [5].
REFERENCES
1. Schein B. M. Relation algebras and function semigroups. Semigroup Forum, 1970, vol. 1, pp. 1-62.
2. Boner F., Poschel F. R. Clones of operations on binary relations. Contributions to General Algebras, 1991, vol. 7, pp. 50-70.
3. Bredikhin D. A. On quasi-identities of algebras of relations with Diophantine operations. Sib. Math. J., 1997, vol. 38, pp. 23-33.
4. Wagner V. V. Restrictiv semigroups. Izv. Vyssh. Uchebn. Zaved. Mat., 1962, no. 6, pp. 19-27 (in Russian).
5. Bredikhin D. A On relation algebras with general superpositions. Colloq. Math. Soc. Janos Bolya, 1994, vol. 54. pp. 11-124.
