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 definition of a hyperdecidable pseudovariety in terms of free profinite 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 finite 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 finite pseudovariety with computable free objects is again hyperdecidable. 1. Introduction A typical problem in...

This article defines the notion of hyperdecidability for a class of finite semigroups, which is closely connected to the notion of decidability. It then proves that the pseudovariety J of J-trivial semigroups is hyperdecidable. 1

Tameness of some locally trivial

Rhodes asked during the Chico conference in 1986 for the calculation of joins of semigroup pseudovarieties. This paper proves that the join JV 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.

kappa-tameness is a strong property of semigroup pseudovarieties related to the membership problem. We prove the kappa-tameness of the following pseudovarieties: N, D, K and LI, which are associated through Eilenberg's correspondence with the varieties of finite and cofinite languages, suffix testable languages, prefix testable languages and factor testable languages respectively.

Pseudovariety joins involving