Profinite categories and semidirect products

Abstract

After developing a theory of implicit operations and proving an analogue of Reiterman's theorem for categories, this paper addresses two complementary questions for semidirect products and two-sided semidirect products of pseudovarieties of semigroups: to determine when a pseudoidentity is valid in it, and to find a basis of pseudoidentities. The first question involves looking into the structure of relatively free profinite objects whereas, for the second question, a general approach is presented which is sufficiently powerful to allow the calculation of many semidirect products. A systematic translation of bases of pseudoidentities of pseudovarieties of categories into characterizations of semidirect products of pseudovarieties of semigroups is given. The latter characterizations, of a syntactic nature, are not effective but may in many cases be reduced to effective characterizations. Several known results are derived as examples — including a syntactic proof of a generalization of the Delay Theorem — and further new applications are obtained using these techniques.