Residually small varieties with modular congruence lattices

Abstract:

We focus on varieties $\mathcal {V}$ of universal algebras whose congruence lattices are all modular. No further conditions are assumed. We prove that if the variety $\mathcal {V}$ is residually small, then the following law holds identically for congruences over algebras in $\mathcal {V}:\beta \cdot [\delta ,\delta ] \leqslant [\beta ,\delta ]$. (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 $\mathcal {V}$ is residually small if and only if it satisfies this commutator identity, and in that case $\mathcal {V}$ 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 $A$.