Atlas: On Identities and Quasiidentities of Semigroups of Relations by Dmitri Bredikhin

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Colloquium on Semigroups
July 17-21, 2000
University of Szeged, Bolyai Institute
Szeged, Hungary

Organizers
Mária B. Szendrei, Eszter K. Horváth, István Szittyai, Géza Takách

On Identities and Quasiidentities of Semigroups of Relations
by
Dmitri Bredikhin
Saratov State Technical University, Saratov, Russia

An involuted semigroup is an algebra (A, ·, ^-1) where (A, ·) is a semigroup and ^-1 is an unary operation which satisfies the identities: (x^-1)^-1=x, (xy)^-1=y^-1x^-1. A binary relation \rho is called difunctional [1] if (x, y) in \rho, (z, y) in \rho, (z, w) in \rho implies (x, w) in \rho . This kind of relations plays important role in modern algebra [2, 3]. Let \Phi be a set of difunctional relations closed under the operations of relation product o and relation inverse ^-1, then (\Phi, o , ^-1) forms an involuted semigroup. Denote by K the class of all involuted semigroups of difunctional relations.

Let L_\infty be a positive first-order logic [4] and L_n be a restriction of L_\infty on formulas with n individual variables. It follows from [5] that all identities (quasiidentities) of involuted semigroups from K can be expressed in L₃. For n >= 3, denote by Eq_n{K} (Qeq_n{K}) the set of all identities (quasiidentities) of involuted semigroups from K that can be derived in L_n, and let V_n{K} (Q_n{K}) be the corresponding variety (quasivariety). Note that V_\infty{K} (Q_\infty{K}) is equal to the variety (quasivariety) generated by K. It is clear that V_n+1{K} subset V_n{K} (Q_n+1{K} subset Q_n{K}) and V_\infty{K} subset V_n{K} (Q_\infty{K} subset Q_n{K}) for any n. It needs four variables to prove that the operation o is associative. For this reason, we suppose that n >= 4.

THEOREM. Q₄(K)=V₄(K)=V_\infty(K); V₄(K) is finitely based; an involuted semigroup (A, ·, ^-1) belongs to V₄(K) if and only if it satisfies the identity xx^-1x=x; Q₅(K) =/= V₅(K).

REFERENCES: [1] Riguet G. Relations binaires, fermetures, correspondences de Galois. Bull. Soc. Math. France, 76(1948), 114-115.

[2] Mac Lane S. An algebra of additive relations. Proc. Nat. Acad. Sci. U.S.A. 5(1961), 1043-1051.

[3] Bredikhin D.A. How can representation theories of inverse semigroups and lattices be united? Semigroup Forum 53(1996), 184-193.

[4] Rasiowa H., Sikorski R. The mathematics of metamathematics. Warszawa, 1963.

[5] Tarski A. On the calculus of relations. J. Symbolic Logic 6(1941), no. 3, 73-89.

Date received: March 21, 2000