Finite axiomatizability of congruence rich varieties (Q1902530)

The notion of a congruence rich variety of algebras is introduced. A variety \(V\) is said to be congruence rich if for each positive integer \(n\) there is a positive integer \(m\) such that every finitely generated algebra in \(V\) with more than \(n\) elements has a homomorphic image with more than \(n\) elements but no more than \(m\) elements. There are examples of varieties of this kind: any finitely generated congruence modular variety and any variety of directoids (introduced in a paper of the first author and \textit{R. Quackenbush} [Algebra Univers. 27, 49-69 (1990; Zbl 0699.08002)]). The question whether locally finite subvarieties of a congruence rich variety are relatively finitely based is discussed. The results obtained are applied to prove that some interesting five-element directoid is not finitely based but fails to be inherently nonfinitely based.