Finite basis theorems for relatively congruence-distributive quasivarieties

Abstract

Q \mathcal {Q} is any quasivariety. A congruence relation Θ \Theta on a member A {\mathbf {A}} of Q \mathcal {Q} is a Q \mathcal {Q} -congruence if A / Θ ∈ Q {\mathbf {A}}/\Theta \in \mathcal {Q} . The set C o n Q A Co{n_\mathcal {Q}}{\mathbf {A}} of all Q \mathcal {Q} -congruences is closed under arbitrary intersection and hence forms a complete lattice C o n Q A {\mathbf {Co}}{{\mathbf {n}}_\mathcal {Q}}{\mathbf {A}} . Q \mathcal {Q} is relatively congruence-distributive if C o n Q A {\mathbf {Co}}{{\mathbf {n}}_\mathcal {Q}}{\mathbf {A}} is distributive for every A ∈ Q {\mathbf {A}} \in \mathcal {Q} . Relatively congruence-distributive quasivarieties occur naturally in the theory of abstract data types. Q \mathcal {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 R \mathcal {R} of an arbitrary quasivariety Q \mathcal {Q} is called a relative subvariety of Q \mathcal {Q} if it is of the form V ∩ Q \mathcal {V} \cap \mathcal {Q} for some variety V \mathcal {V} , i.e., a base for R \mathcal {R} can be obtained by adjoining only identities to a base for Q \mathcal {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.