Hyperdecidable Pseudovarieties and the Calculation of Semidirect Products

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 the deenition of a hyperdecidable pseudovariety in terms of free proonite semigroups, the equivalence with Ash's property in the group case, the behaviour under the operator g of taking the associated global pseudovariety of semigroupoids, and the decidability of V W in case gV is decidable and has a given nite vertex-rank and W is hyperdecidable. A further application of this notion which is given establishes that the join of a hyperdecidable pseudovariety with a locally nite pseudovariety with computable free objects is again hyperdecidable.