IRREDUCIBILITY OF CERTAIN PSEUDOVARIETIES

We prove that the pseudovarieties of all finite semigroups, and of all aperiodic finite semigroups are irreducible for join, for semidirect product and for Mal'cev product. In particular, these pseudovarieties do not admit maximal proper subpseudovarieties. More generally, analog ous results are proved for the pseudovariety of all finite semigroups all of whose subgroups are in a fixed pseudovariety of groups H, provi ded that H is closed under semidirect product.