A finite basis theorem for quasivarieties
A variety V has equationally definable principal meets (EDPM) if there is a finite system of pairs of terms \(<p_ i(x,y,z,w),q_ i(x,y,z,w)>\) for \(i=0,...,n-1\) with the following property: for every algebra \({\mathbb{A}}\in V\) and all a,b,c,d\(\in {\mathbb{A}}\), \[ \Xi (a,b)\cap \Xi (c,d)=\bigvee_{i<n}\Xi (p_ i^{{\mathbb{A}}}(a,b,c,d),q_ i^{{\mathbb{A}}}(a,b,c,d)). \] (\(\Xi\) (a,b) is the principal congruence relation on \({\mathbb{A}}\) generated by a and b.) Examples of varieties with EDPM: Boolean algebras, Heyting algebras, interior algebras, relation algebras, cylindric algebras of finite dimension, regular rings, vector groups, F-rings, vector lattices. The main results of the paper: Theorem 1. Every variety with EDPM is congruence-distributive. Theorem 2. For any variety K the following are equivalent: (i) K has EDPM; (ii) K is congruence-distributive and the finitely generated congruences of K are closed under intersection; (iii) K is congruence-distributive and the class of finitely generated subdirectly irreducible members of K forms a universal class. Theorem 3. If K is any finite subset of a variety with EDPM, then there is a finite base for the quasi-identities of K. Notes: The weaker version of Theorem 2 obtained when condition (i) is replaced by (i') K is congruence-distributive and has EDPM was known. For varieties with EDPM Theorem 3 is a generalization of K. Baker's well known finite basis theorem.
scientific article; zbMATH DE number 3908504