# $F$-inverse monoids as algebraic structures in enriched signature

## TL;DR

This paper studies $F$-inverse monoids within an enriched algebraic signature, characterizing their universal objects and describing free $F$-inverse monoids, thus advancing the algebraic understanding of these structures.

## Contribution

It introduces an enriched signature for $F$-inverse monoids and characterizes universal objects, including free $F$-inverse monoids, in this framework.

## Key findings

- Description of universal objects in $F$-inverse monoids
- Construction of free $F$-inverse monoids for given groups
- Characterization of $F$-inverse monoids as a variety of algebraic structures

## Abstract

Every $F$-inverse monoid can be equipped with the unary operation which maps each element to the maximum element of its $\sigma$-class. In this enriched signature, the class of all $F$-inverse monoids forms a variety of algebraic structures. We describe universal objects in several classes of $F$-inverse monoids, in particular free $F$-inverse monoids. More precisely, for every $X$-generated group $G$ we describe the initial object in the category of all $X$-generated $F$-inverse monoids $F$ for which $F/\sigma=G$.