Bisimple Semigroups
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Relations on Semigroups
International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Relations on Semigroups 1D.D.Padma Priya, 2G.Shobhalatha, 3U.Nagireddy, 4R.Bhuvana Vijaya 1 Sr.Assistant Professor, Department of Mathematics, New Horizon College Of Engineering, Bangalore, India, Research scholar, Department of Mathematics, JNTUA- Anantapuram [email protected] 2Professor, Department of Mathematics, SKU-Anantapuram, India, [email protected] 3Assistant Professor, Rayalaseema University, Kurnool, India, [email protected] 4Associate Professor, Department of Mathematics, JNTUA- Anantapuram, India, [email protected] Abstract: Equivalence relations play a vital role in the study of quotient structures of different algebraic structures. Semigroups being one of the algebraic structures are sets with associative binary operation defined on them. Semigroup theory is one of such subject to determine and analyze equivalence relations in the sense that it could be easily understood. This paper contains the quotient structures of semigroups by extending equivalence relations as congruences. We define different types of relations on the semigroups and prove they are equivalence, partial order, congruence or weakly separative congruence relations. Keywords: Semigroup, binary relation, Equivalence and congruence relations. I. INTRODUCTION [1,2,3 and 4] Algebraic structures play a prominent role in mathematics with wide range of applications in science and engineering. A semigroup -
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. -
8.6 Modular Arithmetic
“mcs” — 2015/5/18 — 1:43 — page 263 — #271 8.6 Modular Arithmetic On the first page of his masterpiece on number theory, Disquisitiones Arithmeticae, Gauss introduced the notion of “congruence.” Now, Gauss is another guy who managed to cough up a half-decent idea every now and then, so let’s take a look at this one. Gauss said that a is congruent to b modulo n iff n .a b/. This is j written a b.mod n/: ⌘ For example: 29 15 .mod 7/ because 7 .29 15/: ⌘ j It’s not useful to allow a modulus n 1, and so we will assume from now on that moduli are greater than 1. There is a close connection between congruences and remainders: Lemma 8.6.1 (Remainder). a b.mod n/ iff rem.a; n/ rem.b; n/: ⌘ D Proof. By the Division Theorem 8.1.4, there exist unique pairs of integers q1;r1 and q2;r2 such that: a q1n r1 D C b q2n r2; D C “mcs” — 2015/5/18 — 1:43 — page 264 — #272 264 Chapter 8 Number Theory where r1;r2 Œ0::n/. Subtracting the second equation from the first gives: 2 a b .q1 q2/n .r1 r2/; D C where r1 r2 is in the interval . n; n/. Now a b.mod n/ if and only if n ⌘ divides the left side of this equation. This is true if and only if n divides the right side, which holds if and only if r1 r2 is a multiple of n. -
On Identities in Groups of Fractions of Cancellative Semigroups
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 7, Pages 1873–1879 S 0002-9939(05)07903-7 Article electronically published on February 24, 2005 ON IDENTITIES IN GROUPS OF FRACTIONS OF CANCELLATIVE SEMIGROUPS S. V. IVANOV AND A. M. STOROZHEV (Communicated by Jonathan I. Hall) Abstract. To solve two problems of Bergman stated in 1981, we construct agroupG such that G contains a free noncyclic subgroup (hence, G satisfies no group identity) and G, as a group, is generated by its subsemigroup that satisfies a nontrivial semigroup identity. 1. Introduction AsemigroupS is called cancellative if for arbitrary a, b, x ∈ S either of equalities xa = xb, ax = bx implies that a = b. AsemigroupS is said to satisfy the left (resp. right) Ore condition if for arbitrary a, b ∈ S there are x, y ∈ S such that xa = yb (resp. ax = by). It is well known that a cancellative semigroup S with the left (or right) Ore condition embeds in a group F(S)=S−1S = SS−1 of its fractions (see [M53], [NT63], [CP67, Theorem 1.23]; recall that, in general, a cancellative semigroup need not embed in a group, see [M37]) and a group F(S)offractionsofS is unique in the sense that if G is a group such that G contains S and G, as a group, is generated by S,thenG is naturally isomorphic to F(S). In particular, if S is a cancellative semigroup satisfying a nontrivial identity, then S satisfies (both) Ore conditions and so S embeds in the group F(S) of its fractions. -
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. -
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. -
Math 327, Solutions to Hour Exam 2
Math 327, Solutions to Hour Exam 2 1. Prove that the function f : N × N ! N defined by f(m; n) = 5m10n is one-to-one. Solution: Begin by observing that we may rewrite the formula for f as f(m; n) = 5m+n2n. Suppose f(m1; n1) = f(m2; n2), then 5m1+n1 2n1 = 5m2+n2 2n2 = k for some common integer k. By the fundamental theorem of arithmetic, the exponents in the prime decomposition of k are unique and hence m1 + n1 = m2 + n2 and n1 = n2: Substituting the second equality into the first we have m1 +n1 = m2 +n1 and hence m1 = m2. Therefore, (m1; n1) = (m2; n2), which shows that the function is one-to-one. 2a. Find all integers x, 0 ≤ x < 10 satisfying the congruence relation 5x ≡ 2 (mod 10). If no such x exists, explain why not. Solution: If the relation has an integer solution x, then there exists an integer k such that 5x − 2 = 10k, which can be rewritten as 2 = 5x − 10k = 5(x − 2k). But since x − 2k is an integer, this shows that 5 2, which is false. This contradiction shows that there are no solutions to the relation. 2b. Find all integers x, 0 ≤ x < 10 satisfying the congruence relation 7x ≡ 6 (mod 10). If no such x exists, explain why not. Solution: By Proposition 4.4.9 in the book, since 7 and 10 are relatively prime, there exists a unique solution to the relation. To find it, we use the Euclidean algorithm to see that 21 = 7(3) + 10(−2), which means that 7(3) ≡ 1(mod 10). -
Arxiv:1707.05940V1 [Math.OA] 19 Jul 2017 ..Aeaiiyo Eirusi Em Fc-Lers66 61 47 C*-Algebras of Terms in Semigroups Red of of Amenability Descriptions C*-Algebra 6.8
SEMIGROUP C*-ALGEBRAS XIN LI Abstract. We give an overview of some recent developments in semigroup C*-algebras. Contents 1. Introduction 2 2. C*-algebrasgeneratedbyleftregularrepresentations 3 3. Examples 4 3.1. The natural numbers 4 3.2. Positive cones in totally ordered groups 4 3.3. Monoids given by presentations 5 3.4. Examples from rings in general, and number theory in particular 8 3.5. Finitely generated abelian cancellative semigroups 9 4. Preliminaries 9 4.1. Embedding semigroups into groups 9 4.2. Graph products 11 4.3. Krull rings 14 5. C*-algebras attached to inverse semigroups, partial dynamical systems, and groupoids 16 5.1. Inverse semigroups 16 5.2. Partial dynamical systems 22 5.3. Etale´ groupoids 26 5.4. The universal groupoid of an inverse semigroup 29 5.5. InversesemigroupC*-algebrasasgroupoidC*-algebras 30 5.6. C*-algebras of partial dynamical systems as C*-algebras of partial transformation groupoids 33 arXiv:1707.05940v1 [math.OA] 19 Jul 2017 5.7. The case of inverse semigroups admitting an idempotent pure partial homomorphism to a group 36 6. Amenability and nuclearity 37 6.1. Groups and groupoids 37 6.2. Amenability for semigroups 40 6.3. Comparing reduced C*-algebras for left cancellative semigroups and their left inverse hulls 42 6.4. C*-algebrasgeneratedbysemigroupsofprojections 47 6.5. The independence condition 54 6.6. Construction of full semigroup C*-algebras 61 6.7. Crossed product and groupoid C*-algebra descriptions of reduced semigroup C*-algebras 63 6.8. Amenability of semigroups in terms of C*-algebras 66 1 2 XIN LI 6.9. -
Von Neumann Regular Cellular Automata
Von Neumann Regular Cellular Automata Alonso Castillo-Ramirez and Maximilien Gadouleau May 29, 2017 Abstract For any group G and any set A, a cellular automaton (CA) is a transformation of the configuration space AG defined via a finite memory set and a local function. Let CA(G; A) be the monoid of all CA over AG. In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element τ ∈ CA(G; A) is von Neumann regular (or simply regular) if there exists σ ∈ CA(G; A) such that τ ◦ σ ◦ τ = τ and σ ◦ τ ◦ σ = σ, where ◦ is the composition of functions. Such an element σ is called a generalised inverse of τ. The monoid CA(G; A) itself is regular if all its elements are regular. We establish that CA(G; A) is regular if and only if |G| = 1 or |A| = 1, and we characterise all regular elements in CA(G; A) when G and A are both finite. Furthermore, we study regular linear CA when A = V is a vector space over a field F; in particular, we show that every regular linear CA is invertible when G is torsion-free elementary amenable (e.g. when G = Zd, d ∈ N) and V = F, and that every linear CA is regular when V is finite-dimensional and G is locally finite with char(F) ∤ o(g) for all g ∈ G. Keywords: Cellular automata, linear cellular automata, monoids, von Neumann regular elements, generalised inverses. 1 Introduction Cellular automata (CA), introduced by John von Neumann and Stanislaw Ulam in the 1940s, are models of computation with important applications to computer science, physics, and theoretical biology. -
UNIT-REGULAR ORTHODOX SEMIGROUPS by R
UNIT-REGULAR ORTHODOX SEMIGROUPS by R. B. McFADDEN (Received 5 May, 1983) Introduction. Unit-regular rings were introduced by Ehrlich [4]. They arose in the search for conditions on a regular ring that are weaker than the ACC, DCC, or finite Goldie dimension, which with von Neumann regularity imply semisimplicity. An account of unit-regular rings, together with a good bibliography, is given by Goodearl [5]. The basic definition of unit-regularity is purely multiplicative; it is simply that for each element x of a monoid S (initially a ring R with identity) there is a unit u of S for which x = xux. The concept of a unit-regular semigroup is a natural one; for example, the full transformation semigroup on a finite set, and the semigroup of endomorphisms of a finite-dimensional vector space, are unit-regular semigroups [1]. Unit-regularity has been studied by Chen and Hsieh [2], by Tirasupa [9], and by McAlister [6]. The connection between unit-regularity and finiteness conditions has been considered by D'Alarcao [3]. The problem of describing the structure of an arbitrary unit-regular semigroup S is difficult. It appears reasonable to attempt to provide such a description in terms of the group of units of S and the set of idempotents of S, and in this direction Blyth and McFadden did determine the structure of a narrow class of unit-regular semigroups. Calling a semigroup S uniquely unit orthodox if it is orthodox and, for each x in S, there exists a unique unit u of S for which x = xux, they proved that every such semigroup is a semidirect product of a group (the group of units of S) and a band (the band of idempotents of S). -
Rees Matrix Covers for Regular Semigroups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 89, 264-279 (1984) Rees Matrix Covers for Regular Semigroups D. B. MCALISTER Department of Mathematics, Northern Iliinois University, De Kalb, Illinois 601 I5 Communicated by G. B. Preston Received June 11, 1982 By a local submonoid, of a regular semigroup S, we mean a subset of the form eSe, where e is an idempotent of S. Many classes of regular semigroups can be defined in terms of properties of their local submonoids. For example, rectangular bands can be characterized as those regular semigroups all of whose local submonoids are trivial while completely simple semigroups are those whose local submonoids are groups. We say that a regular semigroup has a property ‘3 locally, or is a locally 5?7 semigroup, if each local submonoid of S has property %Y. In a previous paper we showed that a regular semigroup is locally inverse if and only if it is an image, by a homomorphism which is one to one on local submonoids, of a regular Rees matrix semigroup over an inverse semigroup. In this paper we extend that result to various other classes of regular semigroups. In particular, we show the analog of this result for locally E-solid semigroups. (A regular semigroup is E-solid if the subsemigroup generated by its idempotents is a union of groups.) The class of locally E-solid regular semigroups is extremely extensive. It includes almost all classes of regular semigroups which have been studied from a structural point of view since inverse semigroups, orthodox semigroups, unions of groups semigroups and their localizations all belong to this class.