Aggregation Operators on Partially Ordered Sets and Their Categorical Foundations
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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. -
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. -
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. -
Endomorphism Monoids of Combinatorial Structures
Endomorphism Monoids of Combinatorial Structures Nik Ruskuc [email protected] School of Mathematics and Statistics, University of St Andrews Dunedin, December 2007 Relational structures Definition I A relational structure X = (X ; Ri (i 2 I )) consists of a set X and a collection of relations defined on X . r I Each relation Ri has an arity ri , meaning Ri ⊆ X i . I The sequence (ri (i 2 I )) is called the signature of X . Combinatorial structures as relational structures Different combinatorial structures can be distinguished by their signatures and special properties required from their relations. Examples I A (simple, undirected) graph is a relational structure with signature (2), such that its only binary relation is symmetric and irreflexive. I A general structure of signature (2) is a digraph. I Posets: signature (2); properties R, AS, T. I Permutations: signature (2,2); properties: two linear orders. Morphisms Definition Let X = (X ; Ri (i 2 I )) be a relational structure. An endomorphism is a mapping θ : X ! X which respects all the relations Ri , i.e. (x1;:::; xk ) 2 Ri ) (x1θ; : : : ; xk θ) 2 Ri : An automorphisms is an invertible endomorphism. Remark For posets and graphs the above becomes: x ≤ y ) xθ ≤ yθ; x ∼ y ) xθ ∼ yθ: Warning: A bijective endomorphism need not be invertible. Take V = Z, E = f(i − 1; i): i ≤ 0g, and f : x 7! x − 1. Morphisms End(X ) = the endomorphism monoid of X Aut(X ) = the automorphism group of X General Problem For a given X , how are X , End(X ) and Aut(X ), and their properties, related? Trans(X ) and Sym(X ) Let E = E(X ) be a trivial relational structure on X . -
Lazy Monoids and Reversible Computation
Imaginary group: lazy monoids and reversible computation J. Gabbay and P. Kropholler REPORT No. 22, 2011/2012, spring ISSN 1103-467X ISRN IML-R- -22-11/12- -SE+spring Imaginary groups: lazy monoids and reversible computation Murdoch J. Gabbay and Peter H. Kropholler Abstract. By constructions in monoid and group theory we exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms, and the category of partially ordered groups and group homomorphisms, such that the unit of the adjunction is in- jective. We also prove a similar result for sets acted on by monoids and groups. We introduce the new notion of lazy homomorphism for a function f be- tween partially-ordered monoids such that f (m m ) f (m) f (m ). ◦ ′ ≤ ◦ ′ Every monoid can be endowed with the discrete partial ordering (m m ≤ ′ if and only if m = m′) so our constructions provide a way of embed- ding monoids into groups. A simple counterexample (the two-element monoid with a non-trivial idempotent) and some calculations show that one can never hope for such an embedding to be a monoid homomor- phism, so the price paid for injecting a monoid into a group is that we must weaken the notion of homomorphism to this new notion of lazy homomorphism. The computational significance of this is that a monoid is an abstract model of computation—or at least of ‘operations’—and similarly a group models reversible computations/operations. By this reading, the adjunc- tion with its injective unit gives a systematic high-level way of faithfully translating an irreversible system to a ‘lazy’ reversible one. -
Math 311 - Introduction to Proof and Abstract Mathematics Group Assignment # 15 Name: Due: at the End of Class on Tuesday, March 26Th
Math 311 - Introduction to Proof and Abstract Mathematics Group Assignment # 15 Name: Due: At the end of class on Tuesday, March 26th More on Functions: Definition 5.1.7: Let A, B, C, and D be sets. Let f : A B and g : C D be functions. Then f = g if: → → A = C and B = D • For all x A, f(x)= g(x). • ∈ Intuitively speaking, this definition tells us that a function is determined by its underlying correspondence, not its specific formula or rule. Another way to think of this is that a function is determined by the set of points that occur on its graph. 1. Give a specific example of two functions that are defined by different rules (or formulas) but that are equal as functions. 2. Consider the functions f(x)= x and g(x)= √x2. Find: (a) A domain for which these functions are equal. (b) A domain for which these functions are not equal. Definition 5.1.9 Let X be a set. The identity function on X is the function IX : X X defined by, for all x X, → ∈ IX (x)= x. 3. Let f(x)= x cos(2πx). Prove that f(x) is the identity function when X = Z but not when X = R. Definition 5.1.10 Let n Z with n 0, and let a0,a1, ,an R such that an = 0. The function p : R R is a ∈ ≥ ··· ∈ 6 n n−1 → polynomial of degree n with real coefficients a0,a1, ,an if for all x R, p(x)= anx +an−1x + +a1x+a0. -
Partial Orders for Representing Uncertainty, Causality, and Decision Making: General Properties, Operations, and Algorithms
Partial Orders are . What We Plan to Do Uncertainty is . Properties of Ordered . Partial Orders for Towards Combining . Representing Uncertainty, Main Result Auxiliary Results Causality, and Decision Proof of the Main Result My Publications Making: General Properties, Home Page Operations, and Algorithms Title Page JJ II Francisco Zapata Department of Computer Science J I University of Texas at El Paso Page1 of 45 500 W. University El Paso, TX 79968, USA Go Back [email protected] Full Screen Close Quit Partial Orders are . What We Plan to Do 1. Partial Orders are Important Uncertainty is . • One of the main objectives of science and engineering Properties of Ordered . is to select the most beneficial decisions. For that: Towards Combining . { we must know people's preferences, Main Result Auxiliary Results { we must have the information about different events Proof of the Main Result (possible consequences of different decisions), and My Publications { since information is never absolutely accurate, we Home Page must have information about uncertainty. Title Page • All these types of information naturally lead to partial orders: JJ II { For preferences, a ≤ b means that b is preferable to J I a. This relation is used in decision theory. Page2 of 45 { For events, a ≤ b means that a can influence b. This Go Back causality relation is used in space-time physics. Full Screen { For uncertain statements, a ≤ b means that a is Close less certain than b (fuzzy logic etc.). Quit Partial Orders are . What We Plan to Do 2. What We Plan to Do Uncertainty is . • In each of the three areas, there is a lot of research Properties of Ordered . -
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. -
Lecture 3 Partially Ordered Sets
Lecture 3 Partially ordered sets Now for something totally different. In life, useful sets are rarely “just” sets. They often come with additional structure. For example, R isn’t useful just because it’s some set. It’s useful because we can add its elements, multiply its elements, and even compare its elements. If A is, for example, the set of all bananas and apples on earth, it’s not as natural to do any of these things with elements of A. Today, we’ll talk about a structure called a partial order. 3.1 Preliminaries First, let me remind you of something called the Cartesian product, or direct product of sets. Definition 3.1.1. Let A and B be sets. Then the Cartesian product of A and B is defined to be the set consisting of all pairs (a, b) where a A and œ b B. We denote the Cartesian product by œ A B. ◊ Example 3.1.2. R2 is the Cartesian product R R. ◊ Example 3.1.3. Let T be the set of all t-shirts in Hiro’s closet, and S the set of all shorts in Hiro’s closet. Then S T represents the collection of all ◊ possible outfits Hiro is willing to consider in August. (I.e., Hiro is willing to consider putting on any of his shorts together with any of his t-shirts.) 23 24 LECTURE 3. PARTIALLY ORDERED SETS Note that we can “repeat” the Cartesian product operation, or apply it many times at once. Example 3.1.4. -
Lecture 19: Alternating Group
MATH 433 Applied Algebra Lecture 19: Alternating group. Abstract groups. Sign of a permutation Theorem 1 For any n ≥ 2 there exists a unique function sgn : S(n) → {−1, 1} such that • sgn(πσ)= sgn(π) sgn(σ) for all π, σ ∈ S(n), • sgn(τ)= −1 for any transposition τ in S(n). A permutation π is called even if it is a product of an even number of transpositions, and odd if it is a product of an odd number of transpositions. It turns out that π is even if sgn(π)=1 and odd if sgn(π)= −1. Theorem 2 (i) sgn(πσ)= sgn(π) sgn(σ) for any π, σ ∈ S(n). (ii) sgn(π−1)= sgn(π) for any π ∈ S(n). (iii) sgn(id) = 1. (iv) sgn(τ)= −1 for any transposition τ. (v) sgn(σ)=(−1)r−1 for any cycle σ of length r. Alternating group Given an integer n ≥ 2, the alternating group on n symbols, denoted An or A(n), is the set of all even permutations in the symmetric group S(n). Theorem (i) For any two permutations π, σ ∈ A(n), the product πσ is also in A(n). (ii) The identity function id is in A(n). (iii) For any permutation π ∈ A(n), the inverse π−1 is in A(n). In other words, the product of even permutations is even, the identity function is an even permutation, and the inverse of an even permutation is even. Theorem The alternating group A(n) has n!/2 elements. -
Theoretical Probability and Statistics
Theoretical Probability and Statistics Charles J. Geyer Copyright 1998, 1999, 2000, 2001, 2008 by Charles J. Geyer September 5, 2008 Contents 1 Finite Probability Models 1 1.1 Probability Mass Functions . 1 1.2 Events and Measures . 5 1.3 Random Variables and Expectation . 6 A Greek Letters 9 B Sets and Functions 11 B.1 Sets . 11 B.2 Intervals . 12 B.3 Functions . 12 C Summary of Brand-Name Distributions 14 C.1 The Discrete Uniform Distribution . 14 C.2 The Bernoulli Distribution . 14 i Chapter 1 Finite Probability Models The fundamental idea in probability theory is a probability model, also called a probability distribution. Probability models can be specified in several differ- ent ways • probability mass function (PMF), • probability density function (PDF), • distribution function (DF), • probability measure, and • function mapping from one probability model to another. We will meet probability mass functions and probability measures in this chap- ter, and will meet others later. The terms “probability model” and “probability distribution” don’t indicate how the model is specified, just that it is specified somehow. 1.1 Probability Mass Functions A probability mass function (PMF) is a function pr Ω −−−−→ R which satisfies the following conditions: its values are nonnegative pr(ω) ≥ 0, ω ∈ Ω (1.1a) and sum to one X pr(ω) = 1. (1.1b) ω∈Ω The domain Ω of the PMF is called the sample space of the probability model. The sample space is required to be nonempty in order that (1.1b) make sense. 1 CHAPTER 1. FINITE PROBABILITY MODELS 2 In this chapter we require all sample spaces to be finite sets. -
Ordered Sets
Ordered Sets Motivation When we build theories in political science, we assume that political agents seek the best element in an appropriate set of feasible alternatives. Voters choose their favorite candidate from those listed on the ballot. Citizens may choose between exit and voice when unsatisfied with their government. Consequently, political science theory requires that we can rank alternatives and identify the best element in various sets of choices. Sets whose elements can be ranked are called ordered sets. They are the subject of this lecture. Relations Given two sets A and B, a binary relation is a subset R ⊂ A × B. We use the notation (a; b) 2 R or more often aRb to denote the relation R holding for an ordered pair (a; b). This is read \a is in the relation R to b." If R ⊂ A × A, we say that R is a relation on A. Example. Let A = fAustin, Des Moines, Harrisburgg and let B = fTexas, Iowa, Pennsylvaniag. Then the relation R = f(Austin, Texas), (Des Moines, Iowa), (Harrisburg, Pennsylvania)g expresses the relation \is the capital of." Example. Let A = fa; b; cg. The relation R = f(a; b); (a; c); (b; c)g expresses the relation \occurs earlier in the alphabet than." We read aRb as \a occurs earlier in the alphabet than b." 1 Properties of Binary Relations A relation R on a nonempty set X is reflexive if xRx for each x 2 X complete if xRy or yRx for all x; y 2 X symmetric if for any x; y 2 X, xRy implies yRx antisymmetric if for any x; y 2 X, xRy and yRx imply x = y transitive if xRy and yRz imply xRz for any x; y; z 2 X Any relation which is reflexive and transitive is called a preorder.