Some Properties of Finitely Decidable Varieties

Let V be a variety whose class of finite members has a decidable first-order theory. We prove that each finite member A of V satisfies the (3,1) and (3,2) transfer principles, and that the minimal sets of prime quotients of type 2 or 3 in A must have empty tails. The first result has already been used by J. Jeong [9] in characterizing the finite subdirectly irreducible members of V with nonabelian monolith. The second result implies that if V is also locally finite and omits type 1, then V is congruence modular.