Pseudovarieties of finite semigroups and applications.

by Jin Mai.Thesis (M.Phil.)--Chinese University of Hong Kong, 1996.Includes bibliographical references (leaves 74-79).Acknowledgement --- p.iAbstract --- p.iiChapter 1. --- Pseudovarieties of finite algebras --- p.1Chapter 2. --- Algebraic automata and formal languages theory --- p.19Chapter 3. --- M-varieties and S-varieties --- p.36Chapter 4. --- The dot-depth hierarchy --- p.48Chapter 5. --- Operators P and P' --- p.62References --- p.7

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

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

On the essential logical structure of inter-universal Teichmüller theory in terms of logical AND “∧”/logical OR “∨” relations: Report on the occasion of the publication of the four main papers on inter-universal Teichmüller theory

The main goal of the present paper is to give a detailed exposition of the essential logical structure of inter-universal Teichmüller theory from the point of view of the Boolean operators --such as the logical AND “∧”logical OR “∨” operators-- of propositional calculus. This essential logical structure of inter-universal Teichmüller theory may be summarized symbolically as follows: A ∧ B = A ∧ (B₁ ∨˙ B₂ ∨˙...) ⇒ A ∧ (B₁∨˙B₂∨˙...∨˙ B́₁ ∨˙ B́₂ ∨˙...) -- where · the “∨˙” denotes the Boolean operator exclusive-OR, i.e., “XOR”; · A, B, B₁, B₂, B́₁, B́₂, denote various propositions; · the logical AND “∧'s” correspond to the Θ-link of inter-universal Teichmüller theory and are closely related to the multiplicative structures of the rings that appear in the domain and codomain of the Θ-link; · the logical XOR “∨˙'s” correspond to various indeterminacies that arise mainly from the log-Kummer-correspondence, i.e., from sequences of iterates of the log-link of inter-universal Teichmüller theory, which may be thought of as a device for constructing additive log-shells. This sort of concatenation of logical AND “∧'s” and logical XOR “∨˙ 's” is reminiscent of the well-known description of the “carry-addition” operation on Teichmüller representatives of the truncated Witt ring ℤ/4ℤ in terms of Boolean addition “∨˙” and Boolean multiplication “∧” in the field F₂ and may be regarded as a sort of “Boolean intertwining” that mirrors, in a remarkable fashion, the “arithmetic intertwining” between addition and multiplication in number fields and local fields, which is, in some sense, the main object of study in inter-universal Teichmüller theory. One important topic in this exposition is the issue of “redundant copies”, i.e., the issue of how the arbitrary identification of copies of isomorphic mathematical objects that appear in the various constructions of inter-universal Teichmüller theory impacts-- and indeed invalidates-- the essential logical structure of inter-universal Teichmüller theory. This issue has been a focal point of fundamental misunderstandings and entirely unnecessary confusion concerning inter-universal Teichmüller theory in certain sectors of the mathematical community. The exposition of the topic of “redundant copies” makes use of many interesting elementary examples from the history of mathematics