Adjoining universal inverses to families of elements of free monoids

TL;DR

This work studies the monoid $M$ obtained by adjoining universal inverses to a (finite or infinite) subset $S$ of the free monoid on $X$. For finite $S$, it constructs an overlap-free set $T$ inverse-equivalent to $S$ and uses the Diamond Lemma to obtain a unique normal form for elements of $X:\mathbf{i}(T)$, with invertible elements forming the free group on $T$ and the rest forming a free product with the free monoid on $X\setminus T$. When $S$ is infinite, $M$ is the direct limit of the finite cases, and a similar normal form exists, though it may not be algorithmically computable from $S$. The paper further examines the structure of these monoids, showing special-case decompositions (e.g., when $T\subseteq X$) and connections to bicyclic monoids, and relates the results to the theory of special monoids, including known decidability results and open questions about iterative constructions and left-right variants.

Abstract

Let $<X>$ be the free monoid on a generating set $X$, and suppose one adjoins to $<X>$ universal 2-sided inverses to a finite set $S$ of its elements. We note an elementary algorithm which yields a normal form for elements of the resulting monoid $M$. We then show that if $S$ is allowed to be infinite, a similar normal form exists, though it cannot necessarily be computed algorithmically. We raise a couple of questions. We note work by others on the related topic of monoids presented by finite families of relations of the form $w = 1$.

Theorems & Definitions (25)

• Lemma 2.7
• proof
• Definition 2.8
• Lemma 2.9
• proof
• Lemma 2.10
• proof
• Definition 3.1
• Proposition 3.2
• proof