Finitely Decidable Congruence Modular Varieties

A class TZ~ of algebras of the same type is said to be finitely decidable iff the first order theory of the class of finite members of 'V is decidable. Let y be a congruence modular variety. In this paper we prove that if "V is finitely decidable, then the following hold. (1) Each finitely generated subvariety of 2^ has a finite bound on the cardinality of its subdirectly irreducible members. (2) Solvable congruences in any locally finite member of Y are abelian. In addition we obtain various necessary conditions on the congruence lattices of finite subdirectly irreducible algebras in "V .