We show that if H is a Fitting pseudovariety of groups and V is a local pseudovariety of monoids, then LH ©m V is local if either V contains the six element Brandt monoid, or H is a non-trivial pseudovariety of groups closed under extension.

A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no non-trivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl∗V has undecidable membership problem. As a consequence, ...

This note introduces the notion of a hyperdecidable pseudovariety. This notion appears naturally in trying to prove decidability of the membership problem of semidirect products of pseudovarieties of semigroups. It turns out to be a generalization of a notion introduced by C. J. Ash in connection with his proof of the \type II" theorem. The main results in this paper include a formulation of th...

J. Rhodes asked during the Chico conference in 1986 for the calculation of joins of semigroup pseudovarieties. This paper proves that the join J∨V of the pseudovariety J of J -trivial semigroups and of any 2-strongly decidable pseudovariety V of completely regular semigroups is decidable. This problem was proposed by the first author for V = G, the pseudovariety of finite groups.

We show that, for every pseudovariety of groups H, the pseudovariety H̄, consisting of all finite semigroups all of whose subgroups lie in H, is irreducible for join and the Mal’cev and semidirect products. The proof involves a Rees matrix construction which motivates the study of iterated Mal’cev products with the pseudovariety of bands. We further provide a strict infinite filtration for H̄ usi...

Let R be the pseudovariety of all finite semigroups whose principal right ideals have a unique generator, let L be its dual, and let J = R∩L. Using the word problem for free pro-J semigroups, it is shown that the bilateral semidirect product Sl ∗∗ J is local, where Sl denotes the pseudovariety of all finite semilattices. The global of the pseudovariety R ∨ L is also computed and it is shown tha...

We give a short proof, using profinite techniques, that idempotent pointlikes, stable pairs and triples are decidable for the pseudovariety of aperiodic monoids. Stable pairs are also described for the pseudovariety of all finite monoids.

In algebraic terms, the insertion of $n$-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality $1\le x^n$. We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity $x^n=1$. In particular, we a...