On the Hyperdecidability of Semidirect Products of Pseudovarieties

The notion of hyperdecidability has been introduced as a tool which is particularly suited for granting decidability of semidirect products. It is shown in this paper that the semidirect product of an hyperdecidable pseudovariety with a pseudovariety whose finitely generated free objects are finite and effectively computable is again hyperdecidable. As instances of this result, one obtains, for example, the hyperdecidability of the pseudovarieties of all finite completely simple semigroups and of all finite bands of left groups. 1. Introduction The semidirect product is one of the most studied algebraic constructions in semigroup theory. In the theory of finite semigroups, applications such as the Krohn-Rhodes complexity have led to the study of the semidirect product as an operation on pseudovarieties and prompted the consideration of the decidability (of the membership) problem for pseudovarieties constructed as semidirect products. Although no example has yet been published, it is ..