Some reducibility properties for pseudovarieties of the form DRH

TL;DR

This paper investigates how the reducibility properties of a group pseudovariety H influence the reducibility of the associated semigroup pseudovariety DRH, focusing on specific classes of equations relevant to decidability.

Contribution

It establishes the relationship between the reducibility of H and DRH for various classes of systems of equations, advancing understanding of their decidability properties.

Findings

Reducibility of H implies reducibility of DRH for certain equations

Connections between group and semigroup pseudovarieties are clarified

Results impact decidability questions in algebraic automata theory

Abstract

Let H be a pseudovariety of groups and DRH be the pseudovariety containing all finite semigroups whose regular R-classes belong to H. We study the relationship between reducibility of H and of DRH with respect to several particular classes of systems of equations. The classes of systems considered (of pointlike, idempotent pointlike and graph equations) are known to play a role in decidability questions concerning pseudovarieties of the forms V * W, V join W, and V malcev W.