A the Green-Rees Local Structure Theory
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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. -
Classes of Semigroups Modulo Green's Relation H
Classes of semigroups modulo Green’s relation H Xavier Mary To cite this version: Xavier Mary. Classes of semigroups modulo Green’s relation H. Semigroup Forum, Springer Verlag, 2014, 88 (3), pp.647-669. 10.1007/s00233-013-9557-9. hal-00679837 HAL Id: hal-00679837 https://hal.archives-ouvertes.fr/hal-00679837 Submitted on 16 Mar 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Classes of semigroups modulo Green’s relation H Xavier Mary∗ Universit´eParis-Ouest Nanterre-La D´efense, Laboratoire Modal’X Keywords generalized inverses; Green’s relations; semigroups 2010 MSC: 15A09, 20M18 Abstract Inverses semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green’s relation H, or in terms of the set of group invertible elements H(S), that allows a study of non-regular semigroups. We then study the interplays between these new classes of semigroups, as well as with known classes of semigroups (notably inverse, orthodox and cryptic semigroups). 1 Introduction The study of special classes of semigroups relies in many cases on properties of the set of idempo- tents, or of regular pairs of elements. -
On the Representation of Inverse Semigroups by Difunctional Relations Nathan Bloomfield University of Arkansas, Fayetteville
University of Arkansas, Fayetteville ScholarWorks@UARK Theses and Dissertations 12-2012 On the Representation of Inverse Semigroups by Difunctional Relations Nathan Bloomfield University of Arkansas, Fayetteville Follow this and additional works at: http://scholarworks.uark.edu/etd Part of the Mathematics Commons Recommended Citation Bloomfield, Nathan, "On the Representation of Inverse Semigroups by Difunctional Relations" (2012). Theses and Dissertations. 629. http://scholarworks.uark.edu/etd/629 This Dissertation is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of ScholarWorks@UARK. For more information, please contact [email protected], [email protected]. ON THE REPRESENTATION OF INVERSE SEMIGROUPS BY DIFUNCTIONAL RELATIONS On the Representation of Inverse Semigroups by Difunctional Relations A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics by Nathan E. Bloomfield Drury University Bachelor of Arts in Mathematics, 2007 University of Arkansas Master of Science in Mathematics, 2011 December 2012 University of Arkansas Abstract A semigroup S is called inverse if for each s 2 S, there exists a unique t 2 S such that sts = s and tst = t. A relation σ ⊆ X × Y is called full if for all x 2 X and y 2 Y there exist x0 2 X and y0 2 Y such that (x; y0) and (x0; y) are in σ, and is called difunctional if σ satisfies the equation σσ-1σ = σ. Inverse semigroups were introduced by Wagner and Preston in 1952 [55] and 1954 [38], respectively, and difunctional relations were introduced by Riguet in 1948 [39]. -
Equidivisible Pseudovarieties of Semigroups
Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 16–05 EQUIDIVISIBLE PSEUDOVARIETIES OF SEMIGROUPS JORGE ALMEIDA AND ALFREDO COSTA Abstract: We give a complete characterization of pseudovarieties of semigroups whose finitely generated relatively free profinite semigroups are equidivisible. Be- sides the pseudovarieties of completely simple semigroups, they are precisely the pseudovarieties that are closed under Mal’cev product on the left by the pseudova- riety of locally trivial semigroups. A further characterization which turns out to be instrumental is as the non-completely simple pseudovarieties that are closed under two-sided Karnofsky-Rhodes expansion. Keywords: semigroup, equidivisible, pseudovariety, Karnofsky-Rhodes expansion, connected expansion, two-sided Cayley graph. AMS Subject Classification (2010): Primary 20M07, 20M05. 1. Introduction A pseudovariety of semigroups is a class of finite semigroups closed un- der taking subsemigroups, homomorphic images and finitary products. In the past few decades, pseudovarieties provided the main framework for the research on finite semigroups, motivated by Eilenberg’s correspondence the- orem between pseudovarieties and varieties of languages. In this context, the finitely generated relatively free profinite semigroups associated to each pseu- dovariety proved to be of fundamental importance. We assume the reader has some familiarity with this background. The books [19, 1] are indicated as supporting references. The paper [2] might also be useful for someone looking for a brief introduction. In this paper we are concerned with equidivisible relatively free profinite semigroups. A semigroup S is equidivisible if for every u,v,x,y ∈ S, the equality uv = xy implies that u = x and v = y, or that there is t ∈ S such that ut = x and v = ty, or such that xt = u and y = tv. -
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 -
3. Inverse Semigroups
3. INVERSE SEMIGROUPS MARK V. LAWSON 1. Introduction Inverse semigroups were introduced in the 1950s by Ehresmann in France, Pre- ston in the UK and Wagner in the Soviet Union as algebraic analogues of pseu- dogroups of transformations. One of the goals of this article is to give some insight into inverse semigroups by showing that they can in fact be seen as extensions of presheaves of groups by pseudogroups of transformations. Inverse semigroups can be viewed as generalizations of groups. Group theory is based on the notion of a symmetry; that is, a structure-preserving bijection. Un- derlying group theory is therefore the notion of a bijection. The set of all bijections from a set X to itself forms a group, S(X), under composition of functions called the symmetric group. Cayley's theorem tells us that each abstract group is isomor- phic to a subgroup of a symmetric group. Inverse semigroup theory, on the other hand, is based on the notion of a partial symmetry; that is, a structure-preserving partial bijection. Underlying inverse semigroup theory, therefore, is the notion of a partial bijection (or partial permutation). The set of all partial bijections from X to itself forms a semigroup, I(X), under composition of partial functions called the symmetric inverse monoid. The Wagner-Preston representation theorem tells us that each abstract inverse semigroup is isomorphic to an inverse subsemigroup of a symmetric inverse monoid. However, symmetric inverse monoids and, by extension, inverse semigroups in general, are endowed with extra structure, as we shall see. The first version of this article was prepared for the Workshop on semigroups and categories held at the University of Ottawa between 2nd and 4th May 2010. -
Uniform Enveloping Semigroupoids for Groupoid Actions
UNIFORM ENVELOPING SEMIGROUPOIDS FOR GROUPOID ACTIONS NIKOLAI EDEKO AND HENRIK KREIDLER Abstract. We establish new characterizations for (pseudo)isometric extensions of topological dynamical systems. For such extensions, we also extend results about relatively invariant measures and Fourier analysis that were previously only known in the minimal case to a significantly larger class, including all transitive systems. To bypass the reliance on minimality of the classical approaches to isometric extensions via the Ellis semigroup, we show that extensions of topological dynamical systems can be described as groupoid actions and then adapt the concept of enveloping semigroups to construct a uniform enveloping semigroupoid for groupoid actions. This approach allows to deal with the more complex orbit structures of nonminimal systems. We study uniform enveloping semigroupoids of general groupoid actions and translate the results back to the special case of extensions of dynamical systems. In particular, we show that, under appropriate assumptions, a groupoid action is (pseudo)isometric if and only if the uniformenvelopingsemigroupoidis actually a compactgroupoid. We also providean operator theoretic characterization based on an abstract Peter–Weyl-type theorem for representations of compact, transitive groupoids on Banach bundles which is of independent interest. Given a topological dynamical system (K,ϕ) consisting of a compact space K and a homeo- morphism ϕ: K → K, its enveloping Ellis semigroup E(K,ϕ) introduced by R. Ellis in [Ell60] is the pointwise closure E(K,ϕ) := {ϕn | n ∈ Z} ⊆ KK . Itisanimportanttoolintopologicaldynamicscapturingthelong-term behaviorofadynamical system. Moreover, it allows to study the system (K,ϕ) via algebraic properties of E(K,ϕ). In particular the elegant theory of compact, right-topological semigroups has been used to describe and study properties of topological dynamical systems. -
On the Homotopy Theory of Monoids
J. Austral. Math. Soc. (Series A) 47 (1989), 171-185 ON THE HOMOTOPY THEORY OF MONOIDS CAROL M. HURWITZ (Received 10 September 1987) Communicated by R. H. Street Abstract In this paper, it is shown that any connected, small category can be embedded in a semi-groupoid (a category in which there is at least one isomorphism between any two elements) in such a way that the embedding includes a homotopy equivalence of classifying spaces. This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category. This construction is essentially algorithmic, and furthermore, yields a finitely presented monoid whenever the small category is finitely presented. Some of these results are generalizations of ideas of McDuff. 1980 Mathematics subject classification [Amer. Math. Soc.) (1985 Revision): 20 M 50, 55 U 35. In 1976, D. M. Kan and W. P. Thurston [5] proved that there is a functorial construction which associates to any path-connected space a discrete group G and a map BG —> X that is a homology equivalence. The kernel of the surjection G — Yl\BG —> YliX is a perfect, normal subgroup P, such that the Quillen ( )+ construction applied to the pair [G, P) yields a space having the same weak homology type as X. Central to their proof is the functorial construction of a homological cone for any discrete group, that is, a natural embedding of a group into an acyclic group. In their construction, G is always uncountable even when X is a finite CW-complex. They posed the following question: given a finite CW-complex X, might one produce a K(G, 1) that is a finite CW-complex homologically equivalent to X. -
OBJ (Application/Pdf)
GROUPOIDS WITH SEMIGROUP OPERATORS AND ADDITIVE ENDOMORPHISM A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS POR THE DEGREE OF MASTER OF SCIENCE BY FRED HENDRIX HUGHES DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA JUNE 1965 TABLE OF CONTENTS Chapter Page I' INTRODUCTION. 1 II GROUPOIDS WITH SEMIGROUP OPERATORS 6 - Ill GROUPOIDS WITH ADDITIVE ENDOMORPHISM 12 BIBLIOGRAPHY 17 ii 4 CHAPTER I INTRODUCTION A set is an undefined termj however, one can say a set is a collec¬ tion of objects according to our sight and perception. Several authors use this definition, a set is a collection of definite distinct objects called elements. This concept is the foundation for mathematics. How¬ ever, it was not until the latter part of the nineteenth century when the concept was formally introduced. From this concept of a set, mathematicians, by placing restrictions on a set, have developed the algebraic structures which we employ. The structures are closely related as the diagram below illustrates. Quasigroup Set The first structure is a groupoid which the writer will discuss the following properties: subgroupiod, antigroupoid, expansive set homor- phism of groupoids in semigroups, groupoid with semigroupoid operators and groupoids with additive endormorphism. Definition 1.1. — A set of elements G - f x,y,z } which is defined by a single-valued binary operation such that x o y ■ z é G (The only restriction is closure) is called a groupiod. Definition 1.2. — The binary operation will be a mapping of the set into itself (AA a direct product.) 1 2 Definition 1.3» — A non-void subset of a groupoid G is called a subgroupoid if and only if AA C A. -
Inverse Monoids in Mathematics &Computer Science
Inverse Semigroups in Mathematics and Computer Science Peter M. Hines Y.C.C.S.A. , Univ. York iCoMET – Sukkur, Pakistan – 2020 [email protected] www.peterhines.info This is not a ‘big-picture’ talk! The overall claim: The structures we find to be deep or elegant in mathematics are often precisely the same as those we consider useful or fundamental in Computer Science – both theoretical and practical. We can only provide evidence for this, one small example at a time. [email protected] www.peterhines.info A small corner of a big picture! We will look at inverse semigroups & monoids: A branch of abstract algebra / semigroup theory. Introduced simultaneously & independently in 1950’s Viktor Wagner (U.S.S.R.) Gordon Preston (U.K.) Theory developed separately, along two different tracks USSR U.S. & Europe with minimal contact between the two sides. Some degree of re-unification in 1990s [email protected] www.peterhines.info A couple of references: [email protected] www.peterhines.info Some -very- basic definitions A semigroup is a set pS; ¨ q with an associative binary operation ¨ : S ˆ S Ñ S. usually written as concatenation: Given a P S and b P S, then ab P S. apbcq “ pabqc for all a; b; c P S. A monoid is a semigroup with an identity1 P S satisfying 1a “ a “ a1 @ a P S A group is a monoid where every a P S has an inverse a´1 P S aa´1 “ 1 “ a´1a @ a P S [email protected] www.peterhines.info Even more basic definitions Simplest examples include free semigroups and monoids. -
CERTAIN FUNDAMENTAL CONGRUENCES on a REGULAR SEMIGROUP! by J
CERTAIN FUNDAMENTAL CONGRUENCES ON A REGULAR SEMIGROUP! by J. M. HOWIE and G. LALLEMENT (Received 21 June, 1965) In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some re- striction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating con- gruence on a regular or inverse semigroup by the authors separately [9, 10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall con- sistently denote by a, P and t] respectively. In § 1 we establish connections between /? and t\ on the one hand and the equivalence relations of Green [7] (see also Clifford and Preston [4, § 2.1]) on the other. If for any relation H on a semigroup S we denote by K* the congruence on S generated by H, then, in the usual notation, In § 2 we show that the intersection of a with jS is the smallest congruence p on S for which Sip is a UBG-semigroup, that is, a band of groups [4, p. -
Green's Relations and Dimension in Abstract Semi-Groups
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 8-1964 Green's Relations and Dimension in Abstract Semi-groups George F. Hampton University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Mathematics Commons Recommended Citation Hampton, George F., "Green's Relations and Dimension in Abstract Semi-groups. " PhD diss., University of Tennessee, 1964. https://trace.tennessee.edu/utk_graddiss/3235 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by George F. Hampton entitled "Green's Relations and Dimension in Abstract Semi-groups." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Don D. Miller, Major Professor We have read this dissertation and recommend its acceptance: Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) July 13, 1962 To the Graduate Council: I am submitting herewith a dissertation written by George Fo Hampton entitled "Green's Relations and Dimension in Abstract Semi groups.-" I recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosop�y, with a major in Mathematics.