A note on the finite basis and finite rank properties for pseudovarieties of semigroups

TL;DR

This paper explores the relationship between finite basis and finite rank properties in pseudovarieties of semigroups, providing new examples of finite rank but not finitely based pseudovarieties, especially in nilpotent cases.

Contribution

It establishes a new result on nilpotent pseudovarieties that generates numerous examples of finite rank pseudovarieties lacking finite bases.

Findings

Many finite rank pseudovarieties are not finitely based.

The paper provides new examples in the context of nilpotent pseudovarieties.

It clarifies the connection between finite basis and finite rank properties.

Abstract

The finite basis property is often connected with the finite rank property, which it entails. Many examples have been produced of finite rank varieties which are not finitely based. In this note, we establish a result on nilpotent pseudovarieties which yields many similar examples in the realm of pseudovarieties of semigroups.