Finite equational bases for finite algebras in a congruence-distributive equational class

AbstractDoes every finite algebraic system A with finitely many operations possess a finite list of polynomial identities (laws), valid in A, from which all other such identities follow? Surprisingly, no (R. C. Lyndon, 1954). The answer is, however, affirmative for various particular kinds of algebraic systems, such as finite groups (Oates and Powell), finite lattices, and even finite lattice-ordered algebraic systems (McKenzie). The purpose of the present paper is to provide a sufficient condition that guarantees an affirmative answer without referring to any particular kind of operation: It is sufficient for A to be a finite member of an equational class of algebraic systems whose congruence lattices are distributive. The proof is constructive. Applications include the case of lattice-ordered algebraic systems