Irreducibility of pseudovarieties of semigroups

We show that, for every pseudovariety of groups H, the pseudovariety consisting of all finite semigroups all of whose subgroups lie in H, is irreducible for join and the Mal'cev and semidirect products. This improves results of Margolis-Sapir-Weil and Rhodes-Steinberg. We further announce some related results on join irreducibility. (Joint work with Ondrej Klíma.)