Pointlike reducibility of pseudovarieties of the form $\bf V*\bf D$

Abstract

In this paper, we investigate the reducibility property of semidirect products of the form $\bf V*\bf D$ relatively to (pointlike) systems of equations of the form $x_1=\cdots=x_n$, where $\bf D$ denotes the pseudovariety of definite semigroups. We establish a connection between pointlike reducibility of $\bf V*\bf D$ and the pointlike reducibility of the pseudovariety $\bf V$. In particular, for the canonical signature $κ$ consisting of the multiplication and the $(ω-1)$-power, we show that $\bf V*\bf D$ is pointlike $κ$-reducible when $\bf V$ is pointlike $κ$-reducible.