Axiomatizable and nonaxiomatizable congruence prevarieties (Q1001459)

For a variety \(V\), \(L(V)\) denotes the class of lattices embeddable in congruence lattices of algebras in \(V\). It is evident that \(L(V)\) is closed under \(I\), \(S\), and \(P\) and hence is a prevariety. If \(V\) satisfies any of the following conditions (a), (b), (c), then \(L(V)\) is first-order axiomatizable: (a) \(V\) is congruence distributive, (b) \(V\) is congruence \(n\)-permutable, (c) \(V\) contains a nontrivial finite strongly soluble algebra. The authors give some criteria for the first-order axiomatizability or nonaxiomatizability of \(L(V)\).