Finite Basis Theorems for Relatively Congruence-distributive Quasivarieties

Q is any quasivariety. A congruence relation 0 on a member A of Q is a Q-congruence if A/0 G Q. The set CouqA. of all Qcongruences is closed under arbitrary intersection and hence forms a complete lattice Cong A. Q. is relatively congruence-distributive if ConjA is distributive for every A e Q.. Relatively congruence-distributive quasivarieties occur naturally in the theory of abstract data types. Q is finitely generated if it is generated by a finite set of finite algebras. The following generalization of Baker's finite basis theorem is proved. Theorem I. Every finitely generated and relatively congruence-distributive quasivariety is finitely based. A subquasivariety JZ of an arbitrary quasivariety Q is called a relative subvariety of Q if it is of the form V n Q for some variety V, i.e., a base for Z can be obtained by adjoining only identities to a base for Q. Theorem II. Every finitely generated relative subvariety of a relatively congruence-distributive quasivariety is finitely based. The quasivariety of generalized equality-test algebras is defined and the structure of its members studied. This gives rise to a finite algebra whose quasi-identities are finitely based while its identities are not. Connections with logic and the algebraic theory of data types are discussed. Introduction. Let Q. be any quasivariety, and let A be any algebra of the same language type as Q (but not necessarily itself a member of fi). A congruence relation © on A is called a Q-congruence if A/6 G Q. The set CouqA of all fi-congruences is closed under arbitrary intersection and hence forms a complete lattice Con^A. Q is relatively congruence-distributive if Cong A is distributive for every A G fi. A subquasivariety R of a quasivariety Q is called a relative subvariety of Q. if it is of the form "V fl fi for some variety "V, i.e., a base for R can be obtained by adjoining only identities to a base for Q. The main purpose of this paper is to prove the following two quasivariety analogues of Baker's finite basis theorem [2]. A quasivariety Q is finitely generated if it is generated by a finite set of finite algebras. Any set of quasi-identities that defines Q is called a base for Q. All algebras are assumed to have only finitely many fundamental operations. THEOREM I. Every finitely generated and relatively congruence-distributive quasivariety is finitely based. THEOREM II. Let Q be a relatively congruence-distributive quasivariety. Then every finitely generated relative subvariety of Q is finitely based. Received by the editors June 16, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 08C15; Secondary 03C05, 08B05, 68Q65.