Proonite Categories and Semidirect Products

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 nd a basis of pseudoidentities. The rst question involves looking into the structure of relatively free proonite objects whereas, for the second question, a general approach is presented which is suuciently powerful to allow the calculation of many semidirect products. A systematic translation of bases of pseudoidenti-ties of pseudovarieties of categories into characterizations of semidirect products of pseudovarieties of semigroups is given. The latter characterizations, of a syntactic nature, are not eeective but may in many cases be reduced to eeective 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.