On Order–Isomorphisms of Stochastic Orders Generated by Partially Ordered Sets with Applications to the Analysis of Chemical Components of Seaweeds
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Chapter 4. Homomorphisms and Isomorphisms of Groups
Chapter 4. Homomorphisms and Isomorphisms of Groups 4.1 Note: We recall the following terminology. Let X and Y be sets. When we say that f is a function or a map from X to Y , written f : X ! Y , we mean that for every x 2 X there exists a unique corresponding element y = f(x) 2 Y . The set X is called the domain of f and the range or image of f is the set Image(f) = f(X) = f(x) x 2 X . For a set A ⊆ X, the image of A under f is the set f(A) = f(a) a 2 A and for a set −1 B ⊆ Y , the inverse image of B under f is the set f (B) = x 2 X f(x) 2 B . For a function f : X ! Y , we say f is one-to-one (written 1 : 1) or injective when for every y 2 Y there exists at most one x 2 X such that y = f(x), we say f is onto or surjective when for every y 2 Y there exists at least one x 2 X such that y = f(x), and we say f is invertible or bijective when f is 1:1 and onto, that is for every y 2 Y there exists a unique x 2 X such that y = f(x). When f is invertible, the inverse of f is the function f −1 : Y ! X defined by f −1(y) = x () y = f(x). For f : X ! Y and g : Y ! Z, the composite g ◦ f : X ! Z is given by (g ◦ f)(x) = g(f(x)). -
Scott Spaces and the Dcpo Category
SCOTT SPACES AND THE DCPO CATEGORY JORDAN BROWN Abstract. Directed-complete partial orders (dcpo’s) arise often in the study of λ-calculus. Here we investigate certain properties of dcpo’s and the Scott spaces they induce. We introduce a new construction which allows for the canonical extension of a partial order to a dcpo and give a proof that the dcpo introduced by Zhao, Xi, and Chen is well-filtered. Contents 1. Introduction 1 2. General Definitions and the Finite Case 2 3. Connectedness of Scott Spaces 5 4. The Categorical Structure of DCPO 6 5. Suprema and the Waybelow Relation 7 6. Hofmann-Mislove Theorem 9 7. Ordinal-Based DCPOs 11 8. Acknowledgments 13 References 13 1. Introduction Directed-complete partially ordered sets (dcpo’s) often arise in the study of λ-calculus. Namely, they are often used to construct models for λ theories. There are several versions of the λ-calculus, all of which attempt to describe the ‘computable’ functions. The first robust descriptions of λ-calculus appeared around the same time as the definition of Turing machines, and Turing’s paper introducing computing machines includes a proof that his computable functions are precisely the λ-definable ones [5] [8]. Though we do not address the λ-calculus directly here, an exposition of certain λ theories and the construction of Scott space models for them can be found in [1]. In these models, computable functions correspond to continuous functions with respect to the Scott topology. It is thus with an eye to the application of topological tools in the study of computability that we investigate the Scott topology. -
The General Linear Group
18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices. -
Generic Filters in Partially Ordered Sets Esfandiar Eslami Iowa State University
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1982 Generic filters in partially ordered sets Esfandiar Eslami Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Eslami, Esfandiar, "Generic filters in partially ordered sets " (1982). Retrospective Theses and Dissertations. 7038. https://lib.dr.iastate.edu/rtd/7038 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 Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. While most advanced technological means to photograph and reproduce this docum. have been used, the quality is heavily dependent upon the quality of the mate submitted. The following explanation of techniques is provided to help you underst markings or notations which may appear on this reproduction. 1.The sign or "target" for pages apparently lacking from the documen photographed is "Missing Page(s)". If it was possible to obtain the missin; page(s) or section, they are spliced into the film along with adjacent pages This may have necessitated cutting through an image and duplicatin; adjacent pages to assure you of complete continuity. 2. When an image on the film is obliterated with a round black mark it is ai indication that the film inspector noticed either blurred copy because o movement during exposure, or duplicate copy. -
Categories, Functors, and Natural Transformations I∗
Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0. -
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. -
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0. -
Homomorphisms and Isomorphisms
Lecture 4.1: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 1 / 13 Motivation Throughout the course, we've said things like: \This group has the same structure as that group." \This group is isomorphic to that group." However, we've never really spelled out the details about what this means. We will study a special type of function between groups, called a homomorphism. An isomorphism is a special type of homomorphism. The Greek roots \homo" and \morph" together mean \same shape." There are two situations where homomorphisms arise: when one group is a subgroup of another; when one group is a quotient of another. The corresponding homomorphisms are called embeddings and quotient maps. Also in this chapter, we will completely classify all finite abelian groups, and get a taste of a few more advanced topics, such as the the four \isomorphism theorems," commutators subgroups, and automorphisms. M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 2 / 13 A motivating example Consider the statement: Z3 < D3. Here is a visual: 0 e 0 7! e f 1 7! r 2 2 1 2 7! r r2f rf r2 r The group D3 contains a size-3 cyclic subgroup hri, which is identical to Z3 in structure only. None of the elements of Z3 (namely 0, 1, 2) are actually in D3. When we say Z3 < D3, we really mean is that the structure of Z3 shows up in D3. -
Friday September 20 Lecture Notes
Friday September 20 Lecture Notes 1 Functors Definition Let C and D be categories. A functor (or covariant) F is a function that assigns each C 2 Obj(C) an object F (C) 2 Obj(D) and to each f : A ! B in C, a morphism F (f): F (A) ! F (B) in D, satisfying: For all A 2 Obj(C), F (1A) = 1FA. Whenever fg is defined, F (fg) = F (f)F (g). e.g. If C is a category, then there exists an identity functor 1C s.t. 1C(C) = C for C 2 Obj(C) and for every morphism f of C, 1C(f) = f. For any category from universal algebra we have \forgetful" functors. e.g. Take F : Grp ! Cat of monoids (·; 1). Then F (G) is a group viewed as a monoid and F (f) is a group homomorphism f viewed as a monoid homomor- phism. e.g. If C is any universal algebra category, then F : C! Sets F (C) is the underlying sets of C F (f) is a morphism e.g. Let C be a category. Take A 2 Obj(C). Then if we define a covariant Hom functor, Hom(A; ): C! Sets, defined by Hom(A; )(B) = Hom(A; B) for all B 2 Obj(C) and f : B ! C, then Hom(A; )(f) : Hom(A; B) ! Hom(A; C) with g 7! fg (we denote Hom(A; ) by f∗). Let us check if f∗ is a functor: Take B 2 Obj(C). Then Hom(A; )(1B) = (1B)∗ : Hom(A; B) ! Hom(A; B) and for g 2 Hom(A; B), (1B)∗(g) = 1Bg = g. -
Math. Res. Lett. 20 (2013), No. 6, 1071–1080 the ISOMORPHISM
Math. Res. Lett. 20 (2013), no. 6, 1071–1080 c International Press 2013 THE ISOMORPHISM RELATION FOR SEPARABLE C*-ALGEBRAS George A. Elliott, Ilijas Farah, Vern I. Paulsen, Christian Rosendal, Andrew S. Toms and Asger Tornquist¨ Abstract. We prove that the isomorphism relation for separable C∗-algebras, the rela- tions of complete and n-isometry for operator spaces, and the relations of unital n-order isomorphisms of operator systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a standard Borel space. 1. Introduction The problem of classifying a collection of objects up to some notion of isomorphism can usually be couched as the study of an analytic equivalence relation on a standard Borel space parametrizing the objects in question. Such relations admit a notion of comparison, Borel reducibility, which allows one to assign a degree of complexity to the classification problem. If X and Y are standard Borel spaces admitting equivalence relations E and F respectively then we say that E is Borel reducible to F , written E ≤B F ,ifthereisaBorelmapΘ:X → Y such that xEy ⇐⇒ Θ(x)F Θ(y). In other words, Θ carries equivalence classes to equivalence classes injectively. We view E as being “less complicated” than F . There are some particularly prominent degrees of complexity in this theory which serve as benchmarks for classification problems in general. For instance, a relation E is classifiable by countable structures (CCS) if it is Borel reducible to the isomorphism relation for countable graphs. Classification problems in functional analysis (our interest here) tend not to be CCS, but may nevertheless be “not too complicated” in that they are Borel reducible to the orbit equivalence relation of a Borel action of a Polish group on a standard Borel space; this property is known as being below a group action. -
PARTIALLY ORDERED TOPOLOGICAL SPACES E(A) = L(A) R\ M(A)
PARTIALLY ORDERED TOPOLOGICAL SPACES L. E. WARD, JR.1 1. Introduction. In this paper we shall consider topological spaces endowed with a reflexive, transitive, binary relation, which, following Birkhoff [l],2 we shall call a quasi order. It will frequently be as- sumed that this quasi order is continuous in an appropriate sense (see §2) as well as being a partial order. Various aspects of spaces possess- ing a quasi order have been investigated by L. Nachbin [6; 7; 8], Birkhoff [l, pp. 38-41, 59-64, 80-82, as well as the papers cited on those pages], and the author [13]. In addition, Wallace [10] has employed an argument using quasi ordered spaces to obtain a fixed point theorem for locally connected continua. This paper is divided into three sections, the first of which is concerned with definitions and preliminary results. A fundamental result due to Wallace, which will have later applications, is proved. §3 is concerned with compact- ness and connectedness properties of chains contained in quasi ordered spaces, and §4 deals with fixed point theorems for quasi ordered spaces. As an application, a new theorem on fixed sets for locally connected continua is proved. This proof leans on a method first employed by Wallace [10]. The author wishes to acknowledge his debt and gratitude to Pro- fessor A. D. Wallace for his patient and encouraging suggestions dur- ing the preparation of this paper. 2. Definitions and preliminary results. By a quasi order on a set X, we mean a reflexive, transitive binary relation, ^. If this relation is also anti-symmetric, it is a partial order. -
Basic Concepts of Linear Order
Basic Concepts of Linear Order Combinatorics for Computer Science (Unit 1) S. Gill Williamson ©S. Gill Williamson 2012 Preface From 1970 to 1990 I ran a graduate seminar on algebraic and algorithmic com- binatorics in the Department of Mathematics, UCSD. From 1972 to 1990 al- gorithmic combinatorics became the principal topic. The seminar notes from 1970 to 1985 were combined and published as a book, Combinatorics for Com- puter Science (CCS), published by Computer Science Press. Each of the "units of study" from the seminar became a chapter in this book. My general goal is to re-create the original presentation of these (largely inde- pendent) units in a form that is convenient for individual selection and study. Here, we isolate Unit 1, corresponding to Chapter 1 of CCS, and reconstruct the original very helpful unit specific index associated with this unit. Theorems, figures, examples, etc., are numbered sequentially: EXERCISE 1.38 and FIGURE 1.62 refer to numbered items 38 and 62 of Unit 1 (or Chap- ter 1 in CCS), etc. CCS contains an extensive bibliography for work prior to 1985. For further references and ongoing research, search the Web, particularly Wikipedia and the mathematics arXiv (arXiv.org). These notes focus on the visualization of algorithms through the use of graph- ical and pictorial methods. This approach is both fun and powerful, preparing you to invent your own algorithms for a wide range of problems. S. Gill Williamson, 2012 http : nwww:cse:ucsd:edun ∼ gill iii iv Table of Contents for Unit 1 Descriptive tools from set theory..............................................................3 relations, equivalence relations, set partitions, image, coimage surjection, injection, bijection, covering relations, Hasse diagrams, exercises and ex- amples.