Finitely decidable congruence modular varieties

Abstract:

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