PROFINITE CATEGORIES AND SEMIDIRECT PRODUCTS

After developing a theory of implicit operations and proving an analog ue of Reiterman's theorem for categories, this paper addresses two com plementary questions for semidirect products and two-sided semidirect products of pseudovarieties of semigroups: to determine when a pseudoi dentity is valid in it, and to find a basis of pseudoidentities. The f irst question involves looking into the structure of relatively free p rofinite objects whereas, for the second question, a general approach is presented which is sufficiently powerful to allow the calculation o f many semidirect products. A systematic translation of bases of pseud oidentities of pseudovarieties of categories into characterizations of semidirect products of pseudovarieties of semigroups is given. The la tter characterizations, of a syntactic nature, are not effective but m ay in many cases be reduced to effective characterizations. Several kn own results are derived as examples - including a syntactic proof of a generalization of the Delay Theorem - and further new applications ar e obtained using these techniques. (C) 1998 Elsevier Science B.V.