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

K. Auinger, G. Kudryavtseva, M. B. Szendrei

Abstract

Every $F$-inverse monoid can be equipped with the unary operation which maps each element to the maximum element of its $σ$-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 $F/σ=G$.

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

Abstract

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

Paper Structure

This paper contains 14 sections, 22 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inverse monoids
  4. Categories of $X$-generated algebraic structures
  5. Cayley graphs of groups
  6. The Margolis--Meakin and the Birget--Rhodes expansions
  7. $F$-inverse monoids in enriched signature
  8. The universal $F$-inverse monoid over an $X$-generated group
  9. Motivation
  10. The model
  11. Universal property
  12. Categorical issues
  13. The variety of perfect $F$-inverse monoids
  14. Additional remark

Key Result

Theorem 2.1

Let $G$ be an $X$-generated group.

Theorems & Definitions (36)

  • Theorem 2.1: MM
  • Proposition 2.2
  • proof
  • Theorem 2.3: Szendrei
  • Proposition 3.1: Kinyon
  • proof
  • Corollary 3.2: Kinyon
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 26 more