Residually small varieties with modular congruence lattices

Abstract

We focus on varieties T of universal algebras whose congruence lattices are all modular. No further conditions are assumed. We prove that if the variety "{ is residually small, then the following law holds identically for congruences over algebras in tV: -[S, S] < [,S]. (The symbols in this formula refer to lattice operations and the commutator operation defined over any modular variety, by Hagemann and Herrmann.) We prove that a finitely generated modular variety "V is residually small if and only if it satisfies this commutator identity, and in that case "i is actually residually < n for some finite integer n. It is further proved that in a modular variety generated by a finite algebra A the chief factors of any finite algebra are bounded in cardinality by the size of A, and every simple algebra in the variety has a cardinality at most that of A.