The word problem for one-relation monoids: a survey

This survey is intended to provide an overview of one of the oldest and most celebrated open problems in combinatorial algebra: the word problem for one-relation monoids. We provide a history of the problem starting in 1914, and give a detailed overview of the proofs of central results, especially those due to Adian and his student Oganesian. After showing how to reduce the problem to the left cancellative case, the second half of the survey focuses on various methods for solving partial cases in this family. We finish with some modern and very recent results pertaining to this problem, including a link to the Collatz conjecture. Along the way, we emphasise and address a number of incorrect and inaccurate statements that have appeared in the literature over the years. We also fill a gap in the proof of a theorem linking special inverse monoids to one-relation monoids, and slightly strengthen the statement of this theorem

String rewriting systems and associated finiteness conditions

We begin with an introduction which describes the thesis in detail, and then a preliminary chapter in which we discuss rewriting systems, associated complexes and finiteness conditions. In particular, we recall the graph of derivations r and the 2- complex V associated to any rewriting system, and the related geometric finiteness conditions F DT and F HT. In §1.4 we give basic definitions and results about finite complete rewriting systems, that is, rewriting systems which rewrite any word in a finite number of steps to its normal form, the unique irreducible word in its congruence class. The main work of the thesis begins in Chapter 2 with some discussion of rewriting systems for groups which are confluent on the congruence class containing the empty word. In §2.1 we characterize groups admitting finite A-complete rewriting systems as those with a A-Dehn presentation, and in §2.2 we give some examples of finite rewriting systems for groups which are A-complete but not complete. For the remainder of the thesis, we study how the properties of finite complete rewriting systems which are discussed in the first chapter are mirrored in higher dimensions. In Chapter 3 we extend the 2-complex V to form a new 3-complex VP, and in Chapter 4 we define new finiteness conditions F DT2 and F HT2 based on the homotopy and homology of this complex. In §4.4 we show that if a monoid admits a finite complete rewriting system, then it is of type F DT2 • The final chapter contains a discussion of alternative ways to define such higher dimensional finiteness conditions. This leads to the introduction, in §5.2, of a variant of the Guba-Sapir homotopy reduction system which can be associated to any co~­ plete rewriting system. This is a rewriting system operating on paths in r, and is complete in the sense that it rewrites paths in a finite number of steps to a unique "normal form"

The word problem and combinatorial methods for groups and semigroups

The subject matter of this thesis is combinatorial semigroup theory. It includes material, in no particular order, from combinatorial and geometric group theory, formal language theory, theoretical computer science, the history of mathematics, formal logic, model theory, graph theory, and decidability theory. In Chapter 1, we will give an overview of the mathematical background required to state the results of the remaining chapters. The only originality therein lies in the exposition of special monoids presented in §1.3, which uni.es the approaches by several authors. In Chapter 2, we introduce some general algebraic and language-theoretic constructions which will be useful in subsequent chapters. As a corollary of these general methods, we recover and generalise a recent result by Brough, Cain & Pfei.er that the class of monoids with context-free word problem is closed under taking free products. In Chapter 3, we study language-theoretic and algebraic properties of special monoids, and completely classify this theory in terms of the group of units. As a result, we generalise the Muller-Schupp theorem to special monoids, and answer a question posed by Zhang in 1992. In Chapter 4, we give a similar treatment to weakly compressible monoids, and characterise their language-theoretic properties. As a corollary, we deduce many new results for one-relation monoids, including solving the rational subset membership problem for many such monoids. We also prove, among many other results, that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. In Chapter 5, we study context-free graphs, and connect the algebraic theory of special monoids with the geometric behaviour of their Cayley graphs. This generalises the geometric aspects of the Muller-Schupp theorem for groups to special monoids. We study the growth rate of special monoids, and prove that a special monoid of intermediate growth is a group