FINITELY DECIDABLE CONGRUENCE MODULAR VARIETIES

A class V of algebras of the same type is said to be finitely decidabl e iff the first order theory of the class of finite members of V is de cidable. Let V be a congruence modular variety. In this paper we prove that if V is finitely decidable, then the following hold. (1) Each fi nitely generated subvariety of V has a finite bound on the cardinality of its subdirectly irreducible members. (2) Solvable congruences in a ny locally finite member of V are abelian. In addition we obtain vario us necessary conditions on the congruence lattices of finite subdirect ly irreducible algebras in V.