On a connection between fuzzy subgroups and $F$-inverse covers of inverse monoids

TL;DR

This paper establishes a categorical connection between fuzzy subgroups and $F$-inverse covers of inverse monoids, showing that fuzzy subgroups can be viewed within the framework of inverse monoid theory.

Contribution

It defines categories for fuzzy subgroups and $F$-inverse covers and proves a full embedding of the former into the latter, linking fuzzy algebra to inverse monoid theory.

Findings

Fuzzy subgroups form a category $rak{F}rak{G}$.

The category of $F$-inverse covers is $rak{F}rak{C}$.

There is a full embedding of $rak{F}rak{G}$ into $rak{F}rak{C}$.

Abstract

We define two categories, the category $\mathfrak{F}\mathfrak{G}$ of fuzzy subgroups, and the category $\mathfrak{F}\mathfrak{C}$ of $F$-inverse covers of inverse monoids, and prove that $\mathfrak{F}\mathfrak{G}$ fully embeds into $\mathfrak{F}\mathfrak{C}$. This shows that, at least from a categorical viewpoint, fuzzy subgroups belong to the standard mathematics as much as they do to the fuzzy one.