Reducibility versus definability for pseudovarieties of semigroups

TL;DR

This paper explores the relationship between reducibility and definability of pseudovarieties of semigroups, providing examples that show the limitations of the converse implication and examining specific cases in language hierarchies.

Contribution

It demonstrates that reducibility does not necessarily imply definability for pseudovarieties, with several counterexamples and a positive case in language hierarchy.

Findings

Reducibility implies definability for the equation x=y.

Counterexamples show the converse does not hold.

A positive example is provided for the inequality x≤ y in a language hierarchy.

Abstract

It is easy to show that a pseudovariety which is reducible with respect to an implicit signature $\sigma$ for the equation $x=y$ can also be defined by $\sigma$-identities. We present several negative examples for the converse using signatures in which the pseudovarieties are usually defined. An ordered example issue from the extended Straubing-Th\'erien hierarchy of regular languages is also shown to provide a positive example for the inequality $x\le y$.