The modularity conjecture holds for linear idempotent varieties

TL;DR

This paper proves the Modularity Conjecture for linear idempotent varieties, showing that the join of two nonmodular varieties remains nonmodular, and explores related properties like n-permutability and congruence identities.

Contribution

It establishes the Modularity Conjecture for linear idempotent varieties and provides new results on n-permutability and congruence identities in this context.

Findings

The Modularity Conjecture holds for linear idempotent varieties.

Join of two nonmodular linear idempotent varieties is nonmodular.

Results on n-permutability and congruence identities in linear varieties.

Abstract

The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning $n$-permutability for some $n$, and the satisfaction of nontrivial congruence identities. Our theorems require a technical result about the equational theory of linear varieties, which might be of independent interest.