Necessary conditions for finite decidability in locally finite varieties admitting strongly abelian behavior

Abstract

We show that several kinds of local behavior in a finite algebra A present obstructions to the decidability of the first-order theory of the finite members of HSP(A). In particular, we show that every solvable congruence in a locally finite, finitely decidable variety is abelian, and that the subdirectly irreducible algebras in such a variety have very constrained congruence geometry, generalizing results of Idziak, Valeriote, and Willard for congruence-modular varieties. We then show that every finitely generated, finitely decidable variety is residually finite (indeed, has a finite residual bound). Finally we modify a construction of Valeriote to give a tighter bound on the essential arity of a sigma-sorted term operation of A.