On $\kappa$-reducibility of pseudovarieties of the form $\mathbf{V}\ast \mathbf{D}$

Abstract

This paper deals with the reducibility property of semidirect products of the form $\mathbf{V}\ast \mathbf{D}$ relatively to graph equation systems, where $\mathbf{D}$ denotes the pseudovariety of definite semigroups. We show that, if the pseudovariety $\mathbf{V}$ is reducible with respect to the canonical signature $\kappa$ consisting of the multiplication and the $(\omega -1)$-power, then $\mathbf{V}\ast \mathbf{D}$ is also reducible with respect to $\kappa$.