On the word problem for special monoids

Abstract

A monoid is called special if it admits a presentation in which all defining relations are of the form $$w = 1$$ w = 1 . Every group is special, but not every monoid is special. In this article, we describe the language-theoretic properties of the word problem, in the sense of Duncan and Gilman, for special monoids in terms of their group of units. We prove that a special monoid has context-free word problem if and only if its group of units is virtually free, giving a full generalisation of the Muller-Schupp theorem. This fully answers, for the class of special monoids, a question posed by Duncan and Gilman (Math Proc Camb Philos Soc 136:513–524, 2004). We describe the congruence classes of words in a special monoid, and prove that these have the same language-theoretic properties as the word problem. This answers a question first posed by Zhang (Math Proc Camb Philos Soc 112:495–505, 1992). As a corollary, we prove that it is decidable (in polynomial time) whether a special one-relation monoid has context-free word problem. This completely answers another question from 1992, also posed by Zhang.

Correction to: The word problem for one-relation monoids: a survey

“…In recent years, there have been some developments on the subject of one-relation monoids. A more complete story of this problem and its history can be found in the author's recent (2021) survey of this problem [55]. However, since then, there have been some new developments on the subject, which bear mentioning here.…”

“…A modern proof, using the language of string rewriting, has been given by Zhang [78]. The author of the present article has also studied special monoids from the point of view of formal language theory in [55].…”

Multiplication Tables and Word-Hyperbolicity in Free Products of Semigroups, Monoids and Groups

“…III]). As proved by the author, a special monoid M has context-free word problem – in the sense of Duncan and Gilman [38, Section 5] – if and only if its group of units is virtually free [91]. Any monoid generated by a finite set A clearly has context-free word problem if and only if it is word-hyperbolic with respect to the regular combing (one direction is trivial by a rational transduction; the other is observed at the beginning of the proof of [25, Theorem 3.1]).…”

“…That is, in the terminology of Section 1.8, we have only used the property that is a reversal-closed super- . (Similarly, general statements involving reversal-closed super- s appear as the main results in previous work by the author [89–91]. ) Hence, the main results of this article (Theorems A, B, 4.5, and their corollaries) remain valid if is replaced in the definition of word-hyperbolicity by any other reversal-closed super- , such as or .…”

“…The terminology monadic ancestor property was introduced by the author in [91], and also appears in [89,90], but was treated implicitly already in [21,73], see especially [73,Lemma 3.4]. The idea of defining classes of languages via ancestry in rewriting systems is not new, and can be traced back at least to, for example, McNaughton et al's Church-Rosser languages [84] or Beaudry et al's McNaughton languages [17,18].…”