DOCSLIB.ORG

The Origin of Pictures: Near-Rings and Algebraic K-Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Origin of Pictures: Near-Rings and Algebraic K-Theory THE ORIGIN OF PICTURES: NEAR-RINGS AND ALGEBRAIC K-THEORY KIYOSHI IGUSA Abstract. These are lecture notes for a talk I gave at the Algebra Seminar at the University of Connecticut on March 21, 2018. I explain the origin of \pictures" which played a large part in many of my earlier papers in topology including my PhD thesis. Since details on the combinatorics of near-rings are already published in my 1982 paper (in which they are called \quasi-rings"), I presented only the ideas and an open question about near-rings which has an important application in topology. Introduction Gordana Todorov, Jerzy Weyman, Kent Orr and I are working on a book about \pic- tures" which have gained renewed attention because they are equivalent to \scattering diagrams". This talk is about very old work that I did to introduce pictures and their re- lation to the cohomology of GL(n; Z). In particular, I will discuss the following pictures (from our book!) and their relationship to H3(GL(n; Z); Z=2). The algebraic side of this story is a fun topic. The cohomology class which detects the \exotic element" of K3Z is the degree 3 class which counts the number of times (modulo 2) that commutativity of addition is used to prove that matrix multiplication is associative! This is an old result (about the obstruction to right distributivity in left near-rings) which I am happy to present in a new light. 1 2 KIYOSHI IGUSA Although I want to avoid the topology, a little bit of history is in order. The focus is the following exact sequence for any group π. f χπ g h π3A(Bπ) −! K3Zπ −! H1(Zπ; Z2π) −! π2A(Bπ) −! K2Zπ ! 0 (1) g; h were defined by A. Hatcher and J. Wagoner [3]. (2) I defined f in my PhD thesis [5]. (3) I showed that elements of K3 of any ring can be represented by 2-dimensional \pictures" [4]. (4) H1(Zπ; Z2π) is Z2π (group ring of π over the field F2 = Z2 with two elements) modulo the conjugation action of π. Thus H1(Zπ; Z2π) = Z2π when π is abelian. (5) This talk is about the mapping χπ. For the trivial group π = 1 we have: χ1 π3A(∗) ! K3Z −! Z2 ! π2A(∗) ! K2Z ! 0 χ1 Z2 ⊕ Z24 ! Z48 −! Z2 ! Z2 ! Z2 ! 0 • Milnor [9]: K2Z = Z2. • Lee-Szczarba [8]: K3Z = Z48. • Waldhausen-Cerf (combining results of [11],[1]): π2A(∗) = Z2. Thus, we already know that χ1 : K3Z ! Z2 is surjective. In [4] I called this map the \Grassmann invariant". This talk has three parts: (1) K3R is the set of deformation classes of \pictures". (2) χπ is the obstruction to matrix associativity for \near-rings". (3) Morse theory on circle bundles gives picture for K3Z having χ1 6= 0 (from [7]). 1. Pictures and H3(G) Definition 1.1. Let G = hX j Yi be a group with generating set X = fx; y; z; · · · g and relation set Y ⊂ F where F is the free group generated by X . Then a picture for G is an oriented planar graph L with (1) Edges labeled with elements of X . (2) Vertices labeled with elements of Y ` Y−1. (3) Base point angle at each vertex. so that the labels of the edges coming to a vertex, read counterclockwise starting at the basepoint angle is the relation at the vertex. Example: THE ORIGIN OF PICTURES: NEAR-RINGS AND ALGEBRAIC K-THEORY 3 y yzx−1 2 Y ` Y−1 * x z Theorem 1.2. Elements of H3(G) are given by \deformation classes" of pictures for G. 3 Example 1.3. For G = Z3 = hx j x i, the generator of H3(Z3) = Z3 is: x x−3 * x3 x * x u Definition 1.4. The Steinberg group of the group ring Zπ has generators xij where −u 1 ≤ i 6= j ≤ n and u 2 π (with inverse written xij ) modulo the relations: u v uv (1)[ xij; xjk] = xik in the word order: u v −u −uv −v xijxjkxij xik xjk u v (2)[ xij; xk`] = 1 when j 6= k and i 6= `. Example 1.5. An example of a picture for the Steinberg group. All arcs oriented clockwise. u x12 xv uv 23 x13 w x14 4 KIYOSHI IGUSA Theorem 1.6 (Gersten). H3(St1(R)) = K3R. Theorem 1.7 (I-Klein). The following picture represents the generator of K3Z = Z48. Proof. Reducing mod 2, this picture and Example 1.3 map to the same element of K3Z2 which is Z3 by [10]. (The homomorphism Z3 ! St(Z2) is given by sending x to x31x13.) So, the 3-torsion part of this element is nonzero. The near-ring associator χ1 : K3Z Z2 is nontrivial on this element. (It is the sum over values at the vertices. Only the blue circled vertex contributes 1.) Therefore, this picture also maps to the generator of the 2-torsion part of Z48. THE ORIGIN OF PICTURES: NEAR-RINGS AND ALGEBRAIC K-THEORY 5 2. Left near-rings Definition 2.1. A left-near-ring (with unity) is structure (R; +; ×; 0; 1) so that (1)( R; +; 0) is a group. (2)( R; ×; 1) is a monoid. (3) (left distributive) a(b + c) = ab + ac () x0 = 0). (4)0 x = 0. In other words, R satisfies all the axioms of a ring with unity except commutativity of addition and right distributivity. In a left near-semi-ring we also drop the axiom of additive inverses. Example: Let M = (M; ×; 1) be a monoid and let G(M) be the free group generated by M. The group law will be written as addition. Thus, the inverse of x 2 M is written −x. G(M) is a left near-ring with multiplication defined as follows. (1) For a; b 2 ±M, ab := (sgn a)(sgn b)jajjbj. Elements of ±M will be called mono- mials. P P (2) If a = aj, b = bk then the product ab is the sum of the monomials ajbk in lexicographic order according to the pair (k; (sgn bk)j). For example, (a1 + a2)(b − c) = (a1 + a2)b + (a1 + a2)(−c) = a1b + a2b −a2c − a1c | {z } reversed We use the notation: (bk) X X ab = ajbk k j 2.1. Right associativity. In general (a1 + a2)b 6= a1b + a2b. These are the sums of the terms aijbk in a different order: (bk) X X X (2.1) a1b + a2b = aijbk i k j (bk) (bk) X X X (2.2) (a1 + a2)b = aijbk k i j The (reduced) obstruction to right distributivity is the symmetric tensor: 2 ρ : G(M) × G(M) × G(M) ! S Z2M = Z2M⊗eZ2M X ρ(a1; a2; b) = aijbk⊗eai0j0 bk0 where the sum is over all pairs of terms which switch order from (2.1) to (2.2). Now compose with the intersection pairing " = h·; ·i : Z2M⊗eZ2M ! Z2M (Con- sider Z2M as the set of all finite subsets of M and take intersection.) Then: X "ρ(a1; a2; b) = haijbk; ai0j0 bk0 i 2 Z2M is the set of elements of M which are commuted with themselves an odd number of times. 6 KIYOSHI IGUSA 2.2. Matrix multiplication. Let Mn(G(M)) be the set of n × n matrices with entries in G(M). If A; B; C 2 Mn(G(M)) we compare A(BC) and (AB)C. Then entries of A(BC) are: X X A(BC)pq = ApiBikCkq k i For (AB)C we have: X (AB)Cpq = (Ap1B1k + Ap2B2k + ··· + ApnBnk) Bkq k To get from one to the other, we need to distribute the product, then permute the terms. We get a (reduced) associator X X α(A; B; C)pq = ρ(ApiBik;ApjBjk; Ckq) + ApiBikCkq⊗eApi0 Bi0k0 Ck0q i<j Take trace followed by the intersection pairing to get " ◦ T r ◦ α(A; B; C) 2 Z2M 3 Theorem 2.2 (I: 1982). This is a 3-cocycle representing a class in H (GLn(ZM); Z2M) 3 which restricts to χM 2 H (Stn(ZM); Z2M) coming from algebraic K-theory. Conjecture 2.3 (Hatcher). For M = π any abelian group, the image of χπ : K3Zπ ! Z2π has order 2. It is enough to prove this for the Klein 4-group π = Z2 × Z2. This significance of this statement is that the cokernel of the map χπ : K3Zπ ! Z2π is a subgroup of π0Diff(M × I rel M × 0) for π1M = π. 3. Calculation The formula for χπ is given as follows. (1) Label each region of the picture L with an element r 2 GLn(Zπ) starting with the unique unbounded region which we label with the identity matrix In and, when we pass a wall from left to right, change the label by right multiplication by the label of the wall: u r " u rx xij ij This is well-defined since the labels around each vertex multiply to In. u v (2) χπ(L) is the sum over all vertices carrying commutators of the form [xij; xik] of the following element of Z2π: X (3.1) rpi husjp; vskpi p where s = (sqp) is the inverse of r = (rpq) and r is the label of any one of the regions adjacent to the vertex. THE ORIGIN OF PICTURES: NEAR-RINGS AND ALGEBRAIC K-THEORY 7 In the example on page 4, the blue circle has x12 coming into the lower right. However, it should be coming into the upper left and, therefore, when drawn properly, x12 should cross x13. So, (i; j; k) = (1; 2; 3).
Load more

Recommended publications
  • On Free Quasigroups and Quasigroup Representations Stefanie Grace Wang Iowa State University
    Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2017 On free quasigroups and quasigroup representations Stefanie Grace Wang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Wang, Stefanie Grace, "On free quasigroups and quasigroup representations" (2017). Graduate Theses and Dissertations. 16298. https://lib.dr.iastate.edu/etd/16298 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On free quasigroups and quasigroup representations by Stefanie Grace Wang A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Jonathan D.H. Smith, Major Professor Jonas Hartwig Justin Peters Yiu Tung Poon Paul Sacks The student author and the program of study committee are solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2017 Copyright c Stefanie Grace Wang, 2017. All rights reserved. ii DEDICATION I would like to dedicate this dissertation to the Integral Liberal Arts Program. The Program changed my life, and I am forever grateful. It is as Aristotle said, \All men by nature desire to know." And Montaigne was certainly correct as well when he said, \There is a plague on Man: his opinion that he knows something." iii TABLE OF CONTENTS LIST OF TABLES .
    [Show full text]
  • Irreducible Representations of Finite Monoids
    U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately.
    [Show full text]
  • Algebras of Boolean Inverse Monoids – Traces and Invariant Means
    C*-algebras of Boolean inverse monoids { traces and invariant means Charles Starling∗ Abstract ∗ To a Boolean inverse monoid S we associate a universal C*-algebra CB(S) and show that it is equal to Exel's tight C*-algebra of S. We then show that any invariant mean on S (in the sense of Kudryavtseva, Lawson, Lenz and Resende) gives rise to a ∗ trace on CB(S), and vice-versa, under a condition on S equivalent to the underlying ∗ groupoid being Hausdorff. Under certain mild conditions, the space of traces of CB(S) is shown to be isomorphic to the space of invariant means of S. We then use many known results about traces of C*-algebras to draw conclusions about invariant means on Boolean inverse monoids; in particular we quote a result of Blackadar to show that any metrizable Choquet simplex arises as the space of invariant means for some AF inverse monoid S. 1 Introduction This article is the continuation of our study of the relationship between inverse semigroups and C*-algebras. An inverse semigroup is a semigroup S for which every element s 2 S has a unique \inverse" s∗ in the sense that ss∗s = s and s∗ss∗ = s∗: An important subsemigroup of any inverse semigroup is its set of idempotents E(S) = fe 2 S j e2 = eg = fs∗s j s 2 Sg. Any set of partial isometries closed under product and arXiv:1605.05189v3 [math.OA] 19 Jul 2016 involution inside a C*-algebra is an inverse semigroup, and its set of idempotents forms a commuting set of projections.
    [Show full text]
  • Binary Opera- Tions, Magmas, Monoids, Groups, Rings, fields and Their Homomorphisms
    1. Introduction In this chapter, I introduce some of the fundamental objects of algbera: binary opera- tions, magmas, monoids, groups, rings, fields and their homomorphisms. 2. Binary Operations Definition 2.1. Let M be a set. A binary operation on M is a function · : M × M ! M often written (x; y) 7! x · y. A pair (M; ·) consisting of a set M and a binary operation · on M is called a magma. Example 2.2. Let M = Z and let + : Z × Z ! Z be the function (x; y) 7! x + y. Then, + is a binary operation and, consequently, (Z; +) is a magma. Example 2.3. Let n be an integer and set Z≥n := fx 2 Z j x ≥ ng. Now suppose n ≥ 0. Then, for x; y 2 Z≥n, x + y 2 Z≥n. Consequently, Z≥n with the operation (x; y) 7! x + y is a magma. In particular, Z+ is a magma under addition. Example 2.4. Let S = f0; 1g. There are 16 = 42 possible binary operations m : S ×S ! S . Therefore, there are 16 possible magmas of the form (S; m). Example 2.5. Let n be a non-negative integer and let · : Z≥n × Z≥n ! Z≥n be the operation (x; y) 7! xy. Then Z≥n is a magma. Similarly, the pair (Z; ·) is a magma (where · : Z×Z ! Z is given by (x; y) 7! xy). Example 2.6. Let M2(R) denote the set of 2 × 2 matrices with real entries. If ! ! a a b b A = 11 12 , and B = 11 12 a21 a22 b21 b22 are two matrices, define ! a b + a b a b + a b A ◦ B = 11 11 12 21 11 12 12 22 : a21b11 + a22b21 a21b12 + a22b22 Then (M2(R); ◦) is a magma.
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • Loop Near-Rings and Unique Decompositions of H-Spaces
    Algebraic & Geometric Topology 16 (2016) 3563–3580 msp Loop near-rings and unique decompositions of H-spaces DAMIR FRANETICˇ PETAR PAVEŠIC´ For every H-space X , the set of homotopy classes ŒX; X possesses a natural al- gebraic structure of a loop near-ring. Albeit one cannot say much about general loop near-rings, it turns out that those that arise from H-spaces are sufficiently close to rings to have a viable Krull–Schmidt type decomposition theory, which is then reflected into decomposition results of H-spaces. In the paper, we develop the algebraic theory of local loop near-rings and derive an algebraic characterization of indecomposable and strongly indecomposable H-spaces. As a consequence, we obtain unique decomposition theorems for products of H-spaces. In particular, we are able to treat certain infinite products of H-spaces, thanks to a recent breakthrough in the Krull–Schmidt theory for infinite products. Finally, we show that indecomposable finite p–local H-spaces are automatically strongly indecomposable, which leads to an easy alternative proof of classical unique decomposition theorems of Wilkerson and Gray. 55P45; 16Y30 Introduction In this paper, we discuss relations between unique decomposition theorems in alge- bra and homotopy theory. Unique decomposition theorems usually state that sum or product decompositions (depending on the category) whose factors are strongly indecomposable are essentially unique. The standard algebraic example is the Krull– Schmidt–Remak–Azumaya theorem. In its modern form, the theorem states that any decomposition of an R–module into a direct sum of indecomposable modules is unique, provided that the endomorphism rings of the summands are local rings; see Facchini [8, Theorem 2.12].
    [Show full text]
  • Monoid Modules and Structured Document Algebra (Extendend Abstract)
    Monoid Modules and Structured Document Algebra (Extendend Abstract) Andreas Zelend Institut fur¨ Informatik, Universitat¨ Augsburg, Germany [email protected] 1 Introduction Feature Oriented Software Development (e.g. [3]) has been established in computer science as a general programming paradigm that provides formalisms, meth- ods, languages, and tools for building maintainable, customisable, and exten- sible software product lines (SPLs) [8]. An SPL is a collection of programs that share a common part, e.g., functionality or code fragments. To encode an SPL, one can use variation points (VPs) in the source code. A VP is a location in a pro- gram whose content, called a fragment, can vary among different members of the SPL. In [2] a Structured Document Algebra (SDA) is used to algebraically de- scribe modules that include VPs and their composition. In [4] we showed that we can reason about SDA in a more general way using a so called relational pre- domain monoid module (RMM). In this paper we present the following extensions and results: an investigation of the structure of transformations, e.g., a condi- tion when transformations commute, insights into the pre-order of modules, and new properties of predomain monoid modules. 2 Structured Document Algebra VPs and Fragments. Let V denote a set of VPs at which fragments may be in- serted and F(V) be the set of fragments which may, among other things, contain VPs from V. Elements of F(V) are denoted by f1, f2,... There are two special elements, a default fragment 0 and an error . An error signals an attempt to assign two or more non-default fragments to the same VP within one module.
    [Show full text]
  • On Cayley Graphs of Algebraic Structures
    On Cayley graphs of algebraic structures Didier Caucal1 1 CNRS, LIGM, France [email protected] Abstract We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative mon- oids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree. 1 Introduction To describe the structure of a group, Cayley introduced in 1878 [9] the concept of graph for any group (G, ·) according to any generating subset S. This is simply the set of labeled s oriented edges g −→ g·s for every g of G and s of S. Such a graph, called Cayley graph, is directed and labeled in S (or an encoding of S by symbols called letters or colors). The study of groups by their Cayley graphs is a main topic of algebraic graph theory [3, 10, 2]. A characterization of unlabeled and undirected Cayley graphs was given by Sabidussi in 1958 [17] : an unlabeled and undirected graph is a Cayley graph if and only if we can find a group with a free and transitive action on the graph. However, this algebraic characterization is not well suited for deciding whether a possibly infinite graph is a Cayley graph. It is pertinent to look for characterizations by graph-theoretic conditions. This approach was clearly stated by Hamkins in 2010: Which graphs are Cayley graphs? [12]. In this paper, we present simple graph-theoretic characterizations of Cayley graphs for firstly left-cancellative and cancellative monoids, and then for groups. These characterizations are then extended to any subset S of left-cancellative magmas, left-quasigroups, quasigroups, and groups.
    [Show full text]
  • Topics in Many-Valued and Quantum Algebraic Logic
    Topics in many-valued and quantum algebraic logic Weiyun Lu August 2016 A Thesis submitted to the School of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science in Mathematics1 c Weiyun Lu, Ottawa, Canada, 2016 1The M.Sc. Program is a joint program with Carleton University, administered by the Ottawa- Carleton Institute of Mathematics and Statistics Abstract Introduced by C.C. Chang in the 1950s, MV algebras are to many-valued (Lukasiewicz) logics what boolean algebras are to two-valued logic. More recently, effect algebras were introduced by physicists to describe quantum logic. In this thesis, we begin by investigating how these two structures, introduced decades apart for wildly different reasons, are intimately related in a mathematically precise way. We survey some connections between MV/effect algebras and more traditional algebraic structures. Then, we look at the categorical structure of effect algebras in depth, and in particular see how the partiality of their operations cause things to be vastly more complicated than their totally defined classical analogues. In the final chapter, we discuss coordinatization of MV algebras and prove some new theorems and construct some new concrete examples, connecting these structures up (requiring a detour through the world of effect algebras!) to boolean inverse semigroups. ii Acknowledgements I would like to thank my advisor Prof. Philip Scott, not only for his guidance, mentorship, and contagious passion which have led me to this point, but also for being a friend with whom I've had many genuinely interesting conversations that may or may not have had anything to do with mathematics.
    [Show full text]
  • 1 Monoids and Groups
    1 Monoids and groups 1.1 Definition. A monoid is a set M together with a map M × M ! M; (x; y) 7! x · y such that (i) (x · y) · z = x · (y · z) 8x; y; z 2 M (associativity); (ii) 9e 2 M such that x · e = e · x = x for all x 2 M (e = the identity element of M). 1.2 Examples. 1) Z with addition of integers (e = 0) 2) Z with multiplication of integers (e = 1) 3) Mn(R) = fthe set of all n × n matrices with coefficients in Rg with ma- trix multiplication (e = I = the identity matrix) 4) U = any set P (U) := fthe set of all subsets of Ug P (U) is a monoid with A · B := A [ B and e = ?. 5) Let U = any set F (U) := fthe set of all functions f : U ! Ug F (U) is a monoid with multiplication given by composition of functions (e = idU = the identity function). 1.3 Definition. A monoid is commutative if x · y = y · x for all x; y 2 M. 1.4 Example. Monoids 1), 2), 4) in 1.2 are commutative; 3), 5) are not. 1 1.5 Note. Associativity implies that for x1; : : : ; xk 2 M the expression x1 · x2 ····· xk has the same value regardless how we place parentheses within it; e.g.: (x1 · x2) · (x3 · x4) = ((x1 · x2) · x3) · x4 = x1 · ((x2 · x3) · x4) etc. 1.6 Note. A monoid has only one identity element: if e; e0 2 M are identity elements then e = e · e0 = e0 1.7 Definition.
    [Show full text]
  • Continuous Monoids and Semirings
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 318 (2004) 355–372 www.elsevier.com/locate/tcs Continuous monoids and semirings Georg Karner∗ Alcatel Austria, Scheydgasse 41, A-1211 Vienna, Austria Received 30 October 2002; received in revised form 11 December 2003; accepted 19 January 2004 Communicated by Z. Esik Abstract Di1erent kinds of monoids and semirings have been deÿned in the literature, all of them named “continuous”. We show their relations. The main technical tools are suitable topologies, among others a variant of the well-known Scott topology for complete partial orders. c 2004 Elsevier B.V. All rights reserved. MSC: 68Q70; 16Y60; 06F05 Keywords: Distributive -algebras; Distributive multioperator monoids; Complete semirings; Continuous semirings; Complete partial orders; Scott topology 1. Introduction Continuous semirings were deÿned by the “French school” (cf. e.g. [16,22]) using a purely algebraic deÿnition. In a recent paper, Esik and Kuich [6] deÿne continuous semirings via the well-known concept of a CPO (complete partial order) and claim that this is a generalisation of the established notion. However, they do not give any proof to substantiate this claim. In a similar vein, Kuich deÿned continuous distributive multioperator monoids (DM-monoids) ÿrst algebraically [17,18] and then, again with Esik, via CPOs. (In the latter paper, DM-monoids are called distributive -algebras.) Also here the relation between the two notions of continuity remains unclear. The present paper clariÿes these issues and discusses also more general concepts. A lot of examples are included.
    [Show full text]
  • Unique Factorization Monoids and Domains
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Number 2, May 1971 UNIQUE FACTORIZATION MONOIDS AND DOMAINS R. E. JOHNSON Abstract. It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M is an ordered monoid and F is a field, the ring ^[[iW"]] of all formal power series with well-ordered support is a uf-domain iff M is naturally ordered (i.e., whenever b <a and aMp\bM¿¿0, then aMQbM). By a monoid, we shall always mean a cancellative semigroup with identity element. All rings are assumed to have unities, and integral domains and fields are not assumed to be commutative. If R is an integral domain, then the multiplicative monoid of nonzero elements is denoted by i?x. For any ring or monoid A, U(A) will denote its group of units. A monoid M is called a unique factorization monoid (uf-monoid) iff for every mEM — U(M), any two factorizations of m have a common refinement. That is, if m=aia2 ■ ■ ■ ar = bib2 ■ ■ ■ bs, then the a's and 6's can be factored, say ai = ci • • • c,, a2 = a+i ■ ■ ■Cj, ■ ■ ■ , 61 = di ■ ■ ■dk, b2=dk+i ■ ■ • dp, ■ ■ ■ , in such a way that cn=dn for all n. An integral domain R is called a unique factorization domain (uf-domain) iff Rx is a uf-monoid. A monoid M with relation < is said to be ordered by < iff < is a transitive linear ordering such that whenever a<b then ac<bc and ca<cb for all cEM.
    [Show full text]