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

TL;DR

This paper studies the reducibility properties of certain semidirect product pseudovarieties, establishing a link between the reducibility of the product and the component pseudovariety, especially under a specific algebraic signature.

Contribution

It demonstrates that the pointlike reducibility of the semidirect product pseudovariety $f V*f D$ can be derived from the pointlike reducibility of $f V$ itself under the canonical signature.

Findings

$f V*f D$ is pointlike $oldsymbol{ m ext{kappa}}$-reducible if $f V$ is

Established a connection between reducibility properties of $f V*f D$ and $f V$

Provided conditions under which reducibility properties are preserved

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 $\kappa$ consisting of the multiplication and the $(\omega-1)$-power, we show that $\bf V*\bf D$ is pointlike $\kappa$-reducible when $\bf V$ is pointlike $\kappa$-reducible.