Nine Chapters on the Semigroup Art
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Computational Techniques in Finite Semigroup Theory
COMPUTATIONAL TECHNIQUES IN FINITE SEMIGROUP THEORY Wilf A. Wilson A Thesis Submitted for the Degree of PhD at the University of St Andrews 2018 Full metadata for this item is available in St Andrews Research Repository at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/16521 This item is protected by original copyright Computational techniques in finite semigroup theory WILF A. WILSON This thesis is submitted in partial fulfilment for the degree of Doctor of Philosophy (PhD) at the University of St Andrews November 2018 Declarations Candidate's declarations I, Wilf A. Wilson, do hereby certify that this thesis, submitted for the degree of PhD, which is approximately 64500 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for any degree. I was admitted as a research student at the University of St Andrews in September 2014. I received funding from an organisation or institution and have acknowledged the funders in the full text of my thesis. Date: . Signature of candidate:. Supervisor's declaration I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of PhD in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. Date: . Signature of supervisor: . Permission for publication In submitting this thesis to the University of St Andrews we understand that we are giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. -
Product Systems Over Ore Monoids
Documenta Math. 1331 Product Systems over Ore Monoids Suliman Albandik and Ralf Meyer Received: August 22, 2015 Revised: October 10, 2015 Communicated by Joachim Cuntz Abstract. We interpret the Cuntz–Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz–Pimsner algebras over Ore monoids are con- structed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz–Pimsner algebra coming from an action of an Ore monoid on a space by topolog- ical correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures. 2010 Mathematics Subject Classification: 46L55, 22A22 Keywords and Phrases: Crossed product; product system; Ore con- ditions; Cuntz–Pimsner algebra; correspondence; groupoid model; higher-rank graph algebra; topological graph algebra. 1. Introduction Let A and B be C∗-algebras. A correspondence from A to B is a Hilbert B-module with a nondegenerate ∗-homomorphism from A to the C∗-algebra of adjointableE operators on . It is called proper if the left A-action is by E compact operators, A K( ). If AB and BC are correspondences from A → E E E to B and from B to C, respectively, then AB B BC is a correspondence from A to C. E ⊗ E A triangle of correspondences consists of three C∗-algebras A, B, C, corre- spondences AB, AC and BC between them, and an isomorphism of corre- E E E spondences u: AB B BC AC ; that is, u is a unitary operator of Hilbert C-modules thatE also⊗ intertwinesE →E the left A-module structures. -
Partial Semigroups and Convolution Algebras
Partial Semigroups and Convolution Algebras Brijesh Dongol, Victor B F Gomes, Ian J Hayes and Georg Struth February 23, 2021 Abstract Partial Semigroups are relevant to the foundations of quantum mechanics and combina- torics as well as to interval and separation logics. Convolution algebras can be understood either as algebras of generalised binary modalities over ternary Kripke frames, in particular over partial semigroups, or as algebras of quantale-valued functions which are equipped with a convolution-style operation of multiplication that is parametrised by a ternary relation. Convolution algebras provide algebraic semantics for various substructural logics, includ- ing categorial, relevance and linear logics, for separation logic and for interval logics; they cover quantitative and qualitative applications. These mathematical components for par- tial semigroups and convolution algebras provide uniform foundations from which models of computation based on relations, program traces or pomsets, and verification components for separation or interval temporal logics can be built with little effort. Contents 1 Introductory Remarks 2 2 Partial Semigroups 3 2.1 Partial Semigroups ................................... 3 2.2 Green’s Preorders and Green’s Relations ....................... 3 2.3 Morphisms ....................................... 4 2.4 Locally Finite Partial Semigroups ........................... 5 2.5 Cancellative Partial Semigroups ............................ 5 2.6 Partial Monoids ..................................... 6 2.7 -
Language and Automata Theory and Applications
LANGUAGE AND AUTOMATA THEORY AND APPLICATIONS Carlos Martín-Vide Characterization • It deals with the description of properties of sequences of symbols • Such an abstract characterization explains the interdisciplinary flavour of the field • The theory grew with the need of formalizing and describing the processes linked with the use of computers and communication devices, but its origins are within mathematical logic and linguistics A bit of history • Early roots in the work of logicians at the beginning of the XXth century: Emil Post, Alonzo Church, Alan Turing Developments motivated by the search for the foundations of the notion of proof in mathematics (Hilbert) • After the II World War: Claude Shannon, Stephen Kleene, John von Neumann Development of computers and telecommunications Interest in exploring the functions of the human brain • Late 50s XXth century: Noam Chomsky Formal methods to describe natural languages • Last decades Molecular biology considers the sequences of molecules formed by genomes as sequences of symbols on the alphabet of basic elements Interest in describing properties like repetitions of occurrences or similarity between sequences Chomsky hierarchy of languages • Finite-state or regular • Context-free • Context-sensitive • Recursively enumerable REG ⊂ CF ⊂ CS ⊂ RE Finite automata: origins • Warren McCulloch & Walter Pitts. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 5:115-133, 1943 • Stephen C. Kleene. Representation of events in nerve nets and -
Dissertation Submitted to the University of Auckland in Fulfillment of the Requirements of the Degree of Doctor of Philosophy (Ph.D.) in Statistics
Thesis Consent Form This thesis may be consulted for the purposes of research or private study provided that due acknowledgement is made where appropriate and that permission is obtained before any material from the thesis is published. Students who do not wish their work to be available for reasons such as pending patents, copyright agreements, or future publication should seek advice from the Graduate Centre as to restricted use or embargo. Author of thesis Jared Tobin Title of thesis Embedded Domain-Specific Languages for Bayesian Modelling and Inference Name of degree Doctor of Philosophy (Statistics) Date Submitted 2017/05/01 Print Format (Tick the boxes that apply) I agree that the University of Auckland Library may make a copy of this thesis available for the ✔ collection of another library on request from that library. ✔ I agree to this thesis being copied for supply to any person in accordance with the provisions of Section 56 of the Copyright Act 1994. Digital Format - PhD theses I certify that a digital copy of my thesis deposited with the University is the same as the final print version of my thesis. Except in the circumstances set out below, no emendation of content has occurred and I recognise that minor variations in formatting may occur as a result of the conversion to digital format. Access to my thesis may be limited for a period of time specified by me at the time of deposit. I understand that if my thesis is available online for public access it can be used for criticism, review, news reporting, research and private study. -
Topology Proceedings COMPACTIFICATIONS of SEMIGROUPS and SEMIGROUP ACTIONS Contents 1. Introduction 2
Submitted to Topology Proceedings COMPACTIFICATIONS OF SEMIGROUPS AND SEMIGROUP ACTIONS MICHAEL MEGRELISHVILI Abstract. An action of a topological semigroup S on X is compacti¯able if this action is a restriction of a jointly contin- uous action of S on a Hausdor® compact space Y . A topo- logical semigroup S is compacti¯able if the left action of S on itself is compacti¯able. It is well known that every Hausdor® topological group is compacti¯able. This result cannot be ex- tended to the class of Tychono® topological monoids. At the same time, several natural constructions lead to compacti¯- able semigroups and actions. We prove that the semigroup C(K; K) of all continuous selfmaps on the Hilbert cube K = [0; 1]! is a universal sec- ond countable compacti¯able semigroup (semigroup version of Uspenskij's theorem). Moreover, the Hilbert cube K under the action of C(K; K) is universal in the realm of all compact- i¯able S-flows X with compacti¯able S where both X and S are second countable. We strengthen some related results of Kocak & Strauss [19] and Ferry & Strauss [13] about Samuel compacti¯cations of semigroups. Some results concern compacti¯cations with sep- arately continuous actions, LMC-compacti¯cations and LMC- functions introduced by Mitchell. Contents 1. Introduction 2 Date: October 7, 2007. 2000 Mathematics Subject Classi¯cation. Primary 54H15; Secondary 54H20. Key words and phrases. semigroup compacti¯cation, LMC-compacti¯cation, matrix coe±cient, enveloping semigroup. 1 2 2. Semigroup actions: natural examples and representations 4 3. S-Compacti¯cations and functions 11 4. -
Semigroup Actions on Sets and the Burnside Ring
SEMIGROUP ACTIONS ON SETS AND THE BURNSIDE RING MEHMET AKIF ERDAL AND ÖZGÜN ÜNLÜ Abstract. In this paper we discuss some enlargements of the category of sets with semi- group actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gröthendieck group. 1. Introduction In the classical terminology, the category of sets with (left) actions of a monoid corresponds to the category of functors from a monoid to the category of sets, by considering monoid as a small category with a single object. If we ignore the identity morphism on the monoid, it corresponds the category of sets with actions of a semigroup, which is conventionally used in applied areas of mathematics such as computer science or physics. For this reason we try to investigate our notions for semigroups, unless we need to use the identity element. In this note we only consider the actions on sets so that we often just write “actions of semigroups" or “actions of monoid" without mentioning “sets". -
Mille Plateaux, You Tarzan: a Musicology of (An Anthropology of (An Anthropology of a Thousand Plateaus))
MILLE PLATEAUX, YOU TARZAN: A MUSICOLOGY OF (AN ANTHROPOLOGY OF (AN ANTHROPOLOGY OF A THOUSAND PLATEAUS)) JOHN RAHN INTRODUCTION: ABOUT TP N THE BEGINNING, or ostensibly, or literally, it was erotic. A Thou- Isand Plateaus (“TP”) evolved from the Anti-Oedipus, also by Gilles Deleuze and Felix Guattari (“D&G”), who were writing in 1968, responding to the mini-revolution in the streets of Paris which catalyzed explosive growth in French thinking, both on the right (Lacan, Girard) and on the left. It is a left-wing theory against patriarchy, and by exten- sion, even against psychic and bodily integration, pro-“schizoanalysis” (Guattari’s métier) and in favor of the Body Without Organs. 82 Perspectives of New Music Eat roots raw. The notion of the rhizome is everywhere: an underground tubercular system or mat of roots, a non-hierarchical network, is the ideal and paradigm. The chapters in TP may be read in any order. The order in which they are numbered and printed cross-cuts the temporal order of the dates each chapter bears (e.g., “November 28, 1947: How Do You Make Yourself a Body Without Organs?”). TP preaches and instantiates a rigorous devotion to the ideal of multiplicity, nonhierarchy, transformation, and escape from boundaries at every moment. TP is concerned with subverting a mindset oriented around an identity which is unchanging essence, but equally subversive of the patriarchal move towards transcendence. This has political implications —as it does in the ultra-right and centrist philosopher Plato, who originally set the terms of debate. Given a choice, though, between one or many Platos, D&G would pick a pack of Platos. -
Jean-Eric Pin and Pascal Weil 1. Introduction
Theoretical Informatics and Applications Theoret. Informatics Appl. 35 (2001) 597–618 A CONJECTURE ON THE CONCATENATION PRODUCT ∗ Jean-Eric Pin1 and Pascal Weil2 Abstract. In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure { this operation corresponds to passing to the upper level in any concatenation hierarchy. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another counterexample, of a different nature, was independently given recently by Steinberg. Taking these two counterex- amples into account, we propose a modified version of our conjecture and some supporting evidence for that new formulation. We show in particular that a solution to our new conjecture would give a solution of the decidability of the levels 2 of the Straubing{Th´erien hierarchy and of the dot-depth hierarchy. Consequences for the other levels are also discussed. Mathematics Subject Classification. 20M07, 68Q45, 20M35. All semigroups and monoids considered in this paper are either finite or free. 1. Introduction The study of the concatenation product goes back to the early years of automata theory. The first major result in this direction was the characterization of star-free languages obtained by Sch¨utzenberger in 1965 [38]. A few years later, Cohen and Brzozowski [11] defined the dot-depth of a star-free language and subdivided the class of star-free languages into Boolean algebras, according to their dot-depth. -
Inverse Semigroups and Varieties of Finite Semigroups
JOURNAL OF ALGEBRA 110, 306323 (1987) inverse Semigroups and Varieties of Finite Semigroups S. W. MARGOLIS Computer Science, Fergvson Budding, Universiry of Nebraska, Lincoln, Nebraska 68588-0115 AND J. E. PIN LITP, 4 Place Jussreu, Tow 55-65, 75252 Paris Cedes 05, France Received November 13, 1984 This paper is the third part of a series of three papers devoted to the study of inverse monoids. It is more specifically dedicated to finite inverse monoids and their connections with formal languages. Therefore, except for free monoids, all monoids considered in this paper will be finite. Throughout the paper we have adopted the point of view of varieties of monoids, which has proved to be an important concept for the study of monoids. Following Eilenberg [2], a variety of monoids is a class of monoids closed under taking submonoids, quotients, and finite direct products. Thus, at first sight, varieties seem to be inadequate for studying inverse monoids since a submonoid of an inverse monoid is not inverse in general. To overcome this difficulty, we consider the variety Inv generated by inverse monoids and closed under taking submonoids, quotients, and finite direct products. Now inverse monoids are simply the regular monoids of this variety and we may use the powerful machinery of variety theory to investigate the algebraic structure of these monoids. Our first result states that Inv is the variety generated by one of the following classes of monoids. (a) Semidirect products of a semilattice by a group. (b) Extensions of a group by a semilattice. (c) Schiitzenberger products of two groups. -
Arxiv:1409.2308V2 [Math.GR] 9 Sep 2014 Hae“Nt Rus Ntette Hs Nld Ok Ntegener the on Th Books with Include Written These Been Title
THE q-THEORY OF FINITE SEMIGROUPS: HISTORY AND MATHEMATICS STUART MARGOLIS The book under review, The q-theory of Finite Semigroups by John Rhodes and Benjamin Steinberg is a remarkable achievement. It com- bines fresh new ideas that give a new light on 40 years of the study of pseudovarieties of finite semigroups and monoids together with the first comprehensive review of the decomposition and complexity theory of finite semigroups since 1976. The authors lay out a new approach that unifies and expands many threads that have run through the study of pseudovarieties and their applications. In addition, a number of fun- damental results about finite semigroups that have never appeared in book form before are included and written from a current point of view. The letter “q” stands for quantum and indeed, the main innovation in the book is to study continuous operators (in the sense of lattice the- ory) on the lattice of pseudovarieties and relational morphisms, giving an analogue to quantum theory. This paper is an expanded version of the short review, [41]. A math- ematics book does not appear independently, but within a historical and mathematical context. Before giving a detailed review of the con- tents of the book, I will give a short history of books in finite semigroup theory and on the theory itself in order to place this book in its proper context. There does not seem to be a traditional way to publish a book review this long. Thus, I am “self publishing” this review. Email feedback to the author from readers would be greatly appreciated and arXiv:1409.2308v2 [math.GR] 9 Sep 2014 I will update the paper when necessary. -
A the Green-Rees Local Structure Theory
A The Green-Rees Local Structure Theory The goal of this appendix is to give an admittedly terse review of the Green- Rees structure theory of stable semigroups (or what might be referred to as the local theory, in comparison with the semilocal theory of Section 4.6). More complete references for this material are [68, 139, 171]. A.1 Ideal Structure and Green's Relations If S is a semigroup, then SI = S I , where I is a newly adjoined identity. If X; Y are subsets of S, then XY[=f gxy x X; y Y . f j 2 2 g Definition A.1.1 (Ideals). Let S be a semigroup. Then: 1. = R S is a right ideal if RS R; ; 6 ⊆ ⊆ 2. = L S is a left ideal if SL L; ; 6 ⊆ ⊆ 3. = I S is an ideal if it is both a left ideal and a right ideal. ; 6 ⊆ If s S, then sSI is the principal right ideal generated by s, SI s is the principal2left ideal generated by s and SI sSI is the principal ideal generated by s. If S is a monoid, then SI s = Ss, sSI = sS and SI sSI = SsS. A semigroup is called left simple, right simple, or simple if it has no proper, respectively, left ideal, right ideal, or ideal. The next proposition is straight- forward; the proof is left to the reader. Proposition A.1.2. Let I; J be ideals of S. Then IJ = ij i I; j J f j 2 2 g is an ideal and = IJ I J.