A Weaker Version of Congruence-Permutability for Semigroup Varieties

Abstract

Congruences α and β are 2.5-permutable if α∨β=αβ∪βα, where ∨ is a union in the congruence lattice and ∪ is the set-theoretic union. A semigroup variety \(\mathcal{V}\) is fi-permutable (fi-2.5-permutable) if every two fully invariant congruences are permutable (2.5-permutable) on all \(\mathcal{V}\)-free semigroups. Previously, a description has been furnished for fi-permutable semigroup varieties. Here, it is proved that a semigroup variety is fi-2.5-permutable iff it either consists of completely simple semigroups, or coincides with a variety of all semilattices, or is contained in one of the explicitly specified nil-semigroup varieties. As a consequence we see that (a) for semigroup varieties that are not nil-varieties, the property of being fi-2.5-permutable is equivalent to being fi-permutable; (b) for a nil-variety \(\mathcal{V}\), if the lattice L(\(\mathcal{V}\)) of its subvarieties is distributive then \(\mathcal{V}\) is fi-2.5-permutable; (c) if \(\mathcal{V}\) is combinatorial or is not completely simple then the fact that \(\mathcal{V}\) is fi-2.5-permutable implies that L(\(\mathcal{V}\)) belongs to a variety generated by a 5-element modular non-distributive lattice.