A Note on the Join of Varieties of Monoids with LI

TL;DR

This paper characterizes the join of certain varieties of finite monoids with the locally trivial semigroups variety using algebraic identities and the concept of essentially-V stamps, connecting algebraic and language-theoretic perspectives.

Contribution

It provides a new algebraic and language-based characterization of the join of monoid varieties with LI, reinterpreting previous topological results from a different angle.

Findings

Characterization of the join of V with LI using identities

Introduction of essentially-V stamps and their equivalence to the join

Conditions on the language varieties corresponding to V

Abstract

In this note, we give a characterisation in terms of identities of the join of $\mathbf{V}$ with the variety of finite locally trivial semigroups $\mathbf{LI}$ for several well-known varieties of finite monoids $\mathbf{V}$ by using classical algebraic-automata-theoretic techniques. To achieve this, we use the new notion of essentially-$\mathbf{V}$ stamps defined by Grosshans, McKenzie and Segoufin and show that it actually coincides with the join of $\mathbf{V}$ and $\mathbf{LI}$ precisely when some natural condition on the variety of languages corresponding to $\mathbf{V}$ is verified.This work is a kind of rediscovery of the work of J. C. Costa around 20 years ago from a rather different angle, since Costa's work relies on the use of advanced developments in profinite topology, whereas what is presented here essentially uses an algebraic, language-based approach.