11 research outputs found
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
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
The B. B. Newman Spelling Theorem
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 high-level 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
MULTIPLICATION TABLES AND WORD-HYPERBOLICITY IN FREE PRODUCTS OF SEMIGROUPS, MONOIDS AND GROUPS
This article studies the properties of word-hyperbolic semigroups and
monoids, i.e. those having context-free multiplication tables with respect to a
regular combing, as defined by Duncan & Gilman. In particular, the preservation
of word-hyperbolicity under taking free products is considered. Under mild
conditions on the semigroups involved, satisfied e.g. by monoids or regular
semigroups, we prove that the semigroup free product of two word-hyperbolic
semigroups is again word-hyperbolic. Analogously, with a mild condition on the
uniqueness of representation for the identity element, satisfied e.g. by
groups, we prove that the monoid free product of two word-hyperbolic monoids is
word-hyperbolic. The methods are language-theoretically general, and apply
equally well to semigroups, monoids, or groups with a
-multiplication table, where is any reversal-closed
super-, in the sense of Greibach. In particular, we deduce
that the free product of two groups with resp. indexed
multiplication tables again has an resp. indexed
multiplication table.Comment: 26(30) pages, 109 references. Comments welcome
Membership problems for positive one-relator groups and one-relation monoids
Motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis and Meakin (2001), we study the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. We give the first known examples of positive one-relator groups with undecidable submonoid membership problem, and apply this to give the first known examples of one-relation monoids with undecidable submonoid membership problem. We construct several infinite families of one-relation monoids with undecidable submonoid membership problem, including examples that are defined by relations of the form but which are not groups, and examples defined by relations of the form where both of and are non-empty. As a consequence we obtain a classification of the right-angled Artin groups that can arise as subgroups of one-relation monoids. We also give examples of monoids with a single defining relation of the form , and examples of the form , with undecidable rational subset membership problem. We give a one-relator group defined by a freely reduced word of the form with positive words, in which the prefix membership problem is undecidable. Finally, we prove the existence of a special two-relator inverse monoid with undecidable word problem, and in which both the relators are positive words. As a corollary, we also find a positive two-relator group with undecidable prefix membership problem. In proving these results, we introduce new methods for proving undecidability of the rational subset membership problem in monoids and groups, including by finding suitable embeddings of certain trace monoids