On the Word Problem for Special Monoids

Abstract

A monoid is special if it admits a presentation in which all the defining relations are of the form w = 1. In this article, we study the word problem, in the sense of Duncan & Gilman, for special monoids, and relate the language-theoretic properties of this set to the word problem for the group of units of the monoid, by developing the theory of the words representing invertible words. We show that a special monoid has regular word problem if and only if it is a finite group. When C is a sufficiently restrictive class of languages, we show that a special monoid has word problem in C if and only if its group of units has word problem in C. As a corollary, we generalise the Muller-Schupp theorem to special monoids: a finitely presented special monoid has context-free word problem if and only if its group of units is virtually free. This completely answers, for the class of special monoids, a question of Duncan & Gilman from 2004. By proving that any context-free monoid has decidable rational subset membership problem, we hence also obtain a large class of special monoids for which this problem is decidable.

On one-relator groups and units of special one-relation inverse monoids

“…Thus we will speak of special inverse monoids and special monoids (and thereby implicitly assume that a presentation of the form given in (2.1) is simultaneously provided). The theory of special monoids is well-developed, originating in work by Adian [2] and Makanin [30,29,47], and developed further by Zhang [70,73,71,72] and Nyberg-Brodda [48], see also [45,Chapters 3 & 5]. Of particular importance for special monoids is the notion of the minimal invertible pieces of the presentation.…”

Non-finitely Generated Maximal Subgroups of Context-free Monoids

“…Inspired by this success, Duncan & Gilman [13, §5] introduced a generalisation of the word problem to finitely generated monoids. This generalisation is not a direct translation; indeed, the language of words equal to the identity element in a monoid is, generally speaking, completely insufficient to describe the structure of the monoid (for notable exceptions to this, see [40]). However, their generalisation, which we shall describe in §2, has the same language-theoretic properties as the original definition in the case that the monoid in question is a group; furthermore, its properties does not generally depend on the finite generating set chosen for the monoid.…”

“…For example, the author has shown that the language-theoretic properties of the word problem of a finitely presented special monoid (cf. §2) are completely determined by its group of units [40]. Here the group of units of a monoid M is the maximal subgroup containing the identity element 1 of M .…”

“…These proofs, written out in full, should also serve to give a taste of the flavour of many of the arguments arising in the theory of special monoids and combinatorial monoid theory. We remark also that some results for finitely presented special monoids fail in the non-finitely presented case; for example, one can show that a finitely presented special monoid M has context-free word problem if and only if its group of units does [40]. This can fail if the assumption of finite presentability is dropped; in fact, there exists a finitely generated special monoid whose word problem is not context-free, and yet having trivial (and hence contextfree) group of units [41].…”

“…Such monoids were introduced by Tseitin [48], and were extensively studied by Adian [2] and Makanin [33,32]. For a more in-depth analysis of special monoids and their history, we refer the reader to [38,40,42,51]. The set {w i | i ∈ I} is called the set of defining words of M .…”