Taylor's modularity conjecture and related problems for idempotent varieties

TL;DR

This paper advances understanding of Taylor's modularity conjecture by establishing new results on the behavior of idempotent varieties and related properties like congruence modularity and cube terms.

Contribution

It provides partial results on Taylor's modularity conjecture, showing how certain properties are preserved or not under interpretability joins and for specific types of varieties.

Findings

Interpretability join of non-congruence modular idempotent varieties is not congruence modular.

Analogous results are proved for idempotent varieties with a cube term.

Results extend to linear varieties and properties like congruence n-permutability.

Abstract

We provide a partial result on Taylor's modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analogue for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence $n$-permutability for a fixed $n$, and satisfying a non-trivial congruence identity.