On $\kappa$-reducibility of pseudovarieties of the form $\bf V*\bf D$

TL;DR

This paper investigates the $oldsymbol{}$-reducibility of pseudovarieties formed by semidirect products with the pseudovariety of definite semigroups, establishing conditions under which reducibility is preserved.

Contribution

It proves that if a pseudovariety $f V$ is $oldsymbol{}$-reducible, then its semidirect product with $f D$ also retains this reducibility property.

Findings

$f V*f D$ is $oldsymbol{}$-reducible if $f V$ is $oldsymbol{}$-reducible.

Reduces the complexity of analyzing semidirect products in algebraic language theory.

Provides a criterion for preserving reducibility in pseudovariety constructions.

Abstract

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