On F-inverse covers of finite-above inverse monoids

TL;DR

This paper establishes a condition involving graph constructions for when finite-above inverse monoids have F-inverse covers via a given group variety, with specific results for Abelian groups.

Contribution

It introduces a new criterion based on graph chains for the existence of F-inverse covers of finite-above inverse monoids, extending understanding of their structure.

Findings

Derived a condition for F-inverse covers involving graph chains.

Provided a simple sufficient condition for non-existence of Abelian F-inverse covers.

Connected the existence of covers to the natural partial order and least group congruence.

Abstract

Finite-above inverse monoids are a common generalization of finite inverse monoids and Margolis--Meakin expansions of groups. Given a finite-above $E$-unitary inverse monoid $M$ and a group variety $\mathit{U}$, we find a condition for $M$ and $\mathit{U}$, involving a construction of descending chains of graphs, which is equivalent to $M$ having an $F$-inverse cover via $\mathit{U}$. In the special case where $\mathit{U}=\mathit{Ab}$, the variety of Abelian groups, we apply this condition to get a simple sufficient condition for $M$ to have no $F$-inverse cover via $\mathit{Ab}$, formulated by means of the natural parial order and the least group congruence of $M$.