Semidirect products of pseudovarieties from the universal algebraist's point of view
The following question is considered. If membership of two pseudovarieties of finite semigroups can be algorithmically determined, does an effective algorithm exist to determine membership of their semidirect product by a given finite semigroup? Of special interest is the case where one of the pseudovarieties consists of all finite semigroups in which every idempotent is a right zero. In tackling such problems the author illustrates the use of some results from \textit{C. J. Ash} [J. Algebra 92, 104-115 (1985; Zbl 0548.08007)] who explored the relationship between varieties, pseudovarieties and generalized varieties. He begins by proving that a generalized variety is locally finite if and only if it is generated by the class of all its finite members. He also characterizes the class of all finite members of the semidirect product of two locally finite generalized varieties as the semidirect product of the two classes of finite algebras obtained from each of the generalized varieties. Using these and other results he obtains characterizations of various semidirect products of pseudovarieties of semigroups.
scientific article; zbMATH DE number 4125704
scientific article