Pointlike reducibility of pseudovarieties of the form $\mathbf{V}\ast \mathbf{D}$

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