Strong solvability and residual finiteness for finitely decidable varieties

TL;DR

This paper proves that finitely generated varieties with decidable first-order theories for finite members are residually finite and have bounded subdirectly irreducible algebras, extending previous results beyond modular congruence lattices.

Contribution

It establishes residual finiteness and bounds on subdirectly irreducible algebras for a broader class of varieties, using properties of strongly solvable radicals.

Findings

Finitely generated varieties with decidable finite member theories are residually finite.

Such varieties have a finite bound on subdirectly irreducible algebra sizes.

The proof involves showing strongly solvable radicals are strongly abelian.

Abstract

If V is a finitely generated variety such that the first-order theory of the finite members of V is decidable, we show that V is residually finite, and in fact has a finite bound on the sizes of subdirectly irreducible algebras. This result generalizes known results which assumed that V has modular congruence lattices. Our proof of the theorem in its full generality proceeds by showing that strongly solvable radicals of algebras in V are strongly abelian.