Three weaker variants of congruence permutability for semigroup varieties

Congruences $\alpha$ and $\beta$ on an algebra $A$ are called $2{.}5$-permutable if the join of $\alpha$ and $\beta$ in the lattice of congruences on $A$ coincides with the set-theoretical union of the relations $\alpha\beta$ and $\beta\alpha$. A semigroup variety $\mathcal V$ is called almost $fi$-permutable [almost weakly $fi$-permutable, almost $fi$-$2{.}5$-permutable] if any two fully invariant congruences on a $\mathcal V$-free object $S$ permute [weakly permute, $2{.}5$-permute] whenever these congruences are contained in the least semilattice congruence on $S$. We completely determine all almost $fi$-permutable varieties, all almost $fi$-$2{.}5$-permutable varieties, and almost weakly $fi$-permutable varieties under the additional assumption that all nilsemigroups in a variety are semigroups with zero multiplication. The first and the third of the corresponding results correct some gaps in two previous papers.