Semidirect product with an order-computable pseudovariety and tameness

Dedicated to the memory of Walter Douglas Munn.The semidirect product of pseudovarieties of semigroups with an ordercomputable pseudovariety is investigated. The essential tool is the natural representation of the corresponding relatively free profinite semigroups and how it transforms implicit signatures. Several results concerning the behavior of the operation with respect to various kinds of tameness properties are obtained as applications.ESF programme “Automata: from Mathematics to Applications (AutoMathA)”.Project PTDC/MAT/65481/2006, which is partly funded by the European Community Fund FEDER.Fundação para a Ciência e a Tecnologia (FCT) through the Centro de Matemática da Universidade do Porto and Centro de Matemática da Universidade do Minh