On the finiteness of certain factorization invariants

Let $H$ be a monoid, $\mathscr F(X)$ be the free monoid on a set $X$, and $\pi_H$ be the unique extension of the identity map on $H$ to a monoid homomorphism $\mathscr F(H) \to H$. Given $A \subseteq H$, an $A$-word $\mathfrak z$ (i.e., an element of $\mathscr F(A)$) is minimal if $\pi_H(\mathfrak z) \ne \pi_H(\mathfrak z')$ for every permutation $\mathfrak z'$ of a proper subword of $\mathfrak z$. The minimal $A$-elasticity of $H$ is then the supremum of all rational numbers $m/n$ with $m, n \in \mathbb N^+$ such that there exist minimal $A$-words $\mathfrak a$ and $\mathfrak b$ of length $m$ and $n$, resp., with $\pi_H(\mathfrak a) = \pi_H(\mathfrak b)$. Among other things, we show that if $H$ is commutative and $A$ is finite, then the minimal $A$-elasticity of $H$ is finite. This yields a non-trivial generalization of the finiteness part of a classical theorem of Anderson et al. from the case where $H$ is cancellative, commutative, and finitely generated (f.g.) modulo units and $A$ is the set $\mathscr A(H)$ of its atoms. We also check that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, f.g. monoid with trivial group of units whose minimal $\mathscr A(H)$-elasticity is infinite.Comment: 13 pages, no figures. To appear in Arkiv f\"or Matemati

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

Using Pictures in Combinatorial Group and Semigroup Theory

Pictures over group presentations are the duals of van Kampen diagrams, known widely in geometric group theory. They have proved to be an effective tool in obtaining results concerning groups. Pictures over semigroup and monoid presentations have recently been introduced and show promise in yielding algebraic information. In Chapter 1 we review existing theory concerning monoid and group presentations, and the concept of pictures over these. We remark that all the monoid presentations considered in this thesis have relations of the form A = B, where A and B are non-empty words. Therefore we refer to the monoid S or semigroup So defined by a monoid presentation. Related to any monoid presentation for a monoid S or semigroup S0, there is a group presentation defining a group G. It is of interest to ascertain exactly how the monoid or semigroup structures are related to the group structure. In Chapter 2 we prove a result which gives sufficient conditions on the group presentation for the embeddability of S in G. We prove that these conditions are implied by the conditions recently given by E.V. Kashintsev. Furthermore, we give an example of a group presentation which satisfies our embeddability conditions but has a corresponding monoid presentation which does not belong to the class of presentations defining embeddable semigroups, studied recently by Cuba. Chapter 3 is concerned with the relationship between conjugacy in S and conjugacy in G. We prove under Kashintsev's embeddablity conditions, and Goldstein and Teymouri's definition of conjugacy in S, that two elements of S which are conjugate in G, are conjugate in S in an 'elementary' way. Chapters 4 and 5 are concerned with relative monoid presentations. Generalising work by Adjan, we introduce the notions of left and right graphs for these presentations. We prove an asphericity result for mixed monoid presentations (for which there exists a natural notion of picture), as well as a cancellation result and an embeddability result for monoids given by relative monoid presentations

Presentations for subsemigroups of groups

This thesis studies subsemigroups of groups from three perspectives: automatic structures, ordinary semigroup presentations, and Malcev presentaions. [A Malcev presentation is a presentation of a special type for a semigroup that can be embedded into a group. A group-embeddable semigroup is Malcev coherent if all of its finitely generated subsemigroups admit finite Malcev presentations.] The theory of synchronous and asynchronous automatic structures for semigroups is expounded, particularly for group-embeddable semigroups. In particular, automatic semigroups embeddable into groups are shown to inherit many of the pleasant geometric properties of automatic groups. It is proved that group- embeddable automatic semigroups admit finite Malcev presentations, and such presentations can be found effectively. An algorithm is exhibited to test whether an automatic semigroup is a free semigroup. Cancellativity of automatic semigroups is proved to be undecidable. Study is made of several classes of groups: virtually free groups; groups that satisfy semigroup laws (in particular [virtually] nilpotent and [virtually] abelian groups); polycyclic groups; free and direct products of certain groups; and one-relator groups. For each of these classes, the question of Malcev coherence is considered, together with the problems of whether finitely generated subsemigroups are finitely presented or automatic. This study yields closure and containment results regarding the class of Malcev coherent groups. The property of having a finite Malcev presentation is shown to be preserved under finite Rees index extensions and subsemigroups. Other concepts of index are also studied

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