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.

We study the language-theoretic aspects of the word problem, in the sense of Duncan & Gilman, of free products of semigroups and monoids. First, we provide algebraic tools for studying classes of languages known as super-AFLs, which generalise e.g. the context-free or the indexed languages. When C is a super-AFL closed under reversal, we prove that the semigroup (monoid) free product of two semigroups (resp. monoids) with word problem in C also has word problem in C. This recovers and generalises a recent result by Brough, Cain & Pfeiffer that the class of context-free semigroups (monoids) is closed under taking free products. As a group-theoretic corollary, we deduce that the word problem of the (group) free product of two groups with word problem in C is also in C. As a particular case, we find that the free product of two groups with indexed word problem has indexed word problem.

A monoid is compressible if it admits a presentation in which all words in the defining relations begin and end with some specified word. Associated to any compressible monoid M is a compressed left monoid L(M ). We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of compressible monoids. We prove that if C is a superAFL, then M has word problem in C if and only if L(M ) has word problem in C. In particular, this applies when C is the class of context-free languages. As a corollary, we deduce that if L(M ) has context-free word problem, then the rational subset membership problem for M is decidable. We also consider subspecial monoids, i.e. one-relation monoids which can be compressed to special monoids. In this case we extend the above results, and show that a subspecial monoid has context-free word problem if and only if all its maximal subgroups are virtually free. This gives a generalisation of the Muller-Schupp theorem to subspecial monoids.

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.

This article aims to be a self-contained account of the history of the B B Newman Spelling Theorem, including the historical context in which it arose. First, an account of B B Newman and how he came to prove his Spelling Theorem is given, together with a description of the author's efforts to track this information down. Following this, a highlevel description of combinatorial group theory is given. This is then tied in with a description of the history of the word problem, a fundamental problem in the area. After a description of some of the theory of one-relator groups, an important part of combinatorial group theory, the natural division line into the torsion and torsion-free case for such groups is described. This culminates in a statement of and general discussion about the B B Newman Spelling Theorem and its importance.