Completely regular semigroup varieties whose free objects have weakly permutable fully invariant congruences. (Q1434110)

Congruences \(\alpha,\beta\) on an algebra `weakly permute' if \(\alpha\beta\alpha=\beta\alpha\beta\). A semigroup variety is called `weakly fi-permutable' if the fully invariant congruences on any of its relatively free objects are weakly permutable. Similar definitions apply for permutability. The author proves that any weakly fi-permutable semigroup variety consists either of completely regular semigroups (unions of groups) or of nilsemigroups. The latter case will be discussed in a later paper by the author. In this paper, he shows that a completely regular semigroup variety is weakly fi-permutable if and only if it consists either of completely simple semigroups or of semilattices of groups. (As the author notes, completely simple semigroup varieties are in fact fi-permutable. More generally, for any completely regular variety, the fully invariant congruences that are contained in the least semilattice congruence on any of its relatively free semigroups are permutable.)