A Weaker Version of Congruence-Permutability for Semigroup Varieties

Congruences $\alpha$ and $\beta$ are 2.5-permutable if $\alpha\vee\beta=\alpha\beta\cup\beta\alpha$, where $\vee$ is a union in the congruence lattice and $\cup$ 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 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.