Two weaker variants of congruence permutability for monoid varieties

TL;DR

This paper characterizes specific monoid varieties where certain congruences permute and discovers new varieties with distributive subvariety lattices, advancing understanding of monoid structure.

Contribution

It fully classifies monoid varieties with permuting fully invariant congruences and identifies new varieties with distributive subvariety lattices.

Findings

Complete classification of monoid varieties with permuting fully invariant congruences.

Discovery of several new monoid varieties with distributive subvariety lattices.

Abstract

We completely determine all varieties of monoids on whose free objects all fully invariant congruences or all fully invariant congruences contained in the least semilattice congruence permute. Along the way, we find several new monoid varieties with the distributive subvariety lattice (only a few examples of varieties with such a property are known so far).