Lazy Monoids and Reversible Computation
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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. -
ORDERED GROUPS VM Kopytov UDC 512.545
ORDERED GROUPS V. M. Kopytov UDC 512.545 In the paper "Ordered groups" there is given a survey of papers on ordered groups, reviewed in RZh Mathematika in 1975-1980. Starting in 1963 there occurred a qualitative jump in the theory of ordered groups, evoked by the inten- sive investigation of linerarly ordered (l. o. ) groups and the development of the theory of groups of automor- phisms of I.o. sets. As a result many sections of the theory of lattice ordered groups (l-groups) acquired an orderly and organized form, and profound classificationalresults were obtained in them. A natural con- sequence of this was the appearance in the recent past of several monographs on the theory of ordered groups, in particular, the books of Kokorin and Kopytov [26] (English translation [154]), Mura and Rhemtulla [182], Bigard, Keimel, and Wolfenstein [78]. The present survey is written on the materials of the Ref. Zh. "Matematika," mainly for the years 1975-1980, and reflects practicallyall directions of development of the theory of ordered groups with some accent on linearly and lattice ordered groups, which is explained by the intensivity and diversity of the inves- tigations in these domains. In the survey there are included some results reviewed in the years 1970-1974 and not reflected in previous surveys of the collection "Algebra. Topology. Geometry" of the annual 'Itogi Nauki i Tekhniki" [12, 13] or the book [26]. i. Linearly Ordered Groups Investigations on 1. o. groups were carried out mainly in the directions designated at the end of the six- ties and formulated in [26]. -
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. -
Weakly Abelian Lattice-Ordered Groups 1
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 3, Pages 677{684 S 0002-9939(00)05706-3 Article electronically published on September 20, 2000 WEAKLY ABELIAN LATTICE-ORDERED GROUPS A. M. W. GLASS (Communicated by Stephen D. Smith) Respectfully dedicated (with gratitude) to W. Charles Holland on his 65th Birthday Abstract. Every nilpotent lattice-ordered group is weakly Abelian; i.e., sat- isfies the identity x−1(y _ 1)x _ (y _ 1)2 =(y _ 1)2. In 1984, V. M. Kopytov asked if every weakly Abelian lattice-ordered group belongs to the variety generated by all nilpotent lattice-ordered groups [The Black Swamp Prob- lem Book, Question 40]. In the past 15 years, all attempts have centred on finding counterexamples. We show that two constructions of weakly Abelian lattice-ordered groups fail to be counterexamples. They include all preiously considered potential counterexamples and also many weakly Abelian ordered free groups on finitely many generators. If every weakly Abelian ordered free group on finitely many generators belongs to the variety generated by all nilpo- tent lattice-ordered groups, then every weakly Abelian lattice-ordered group belongs to this variety. This paper therefore redresses the balance and suggests that Kopytov's problem is even more intriguing. 1. Basic definitions and facts A group equipped with a partial order ≤ is called a p.o.group if axb ≤ ayb whenever x ≤ y. If, further, the partial order is a lattice (any pair of elements x & y have a least upper bound (denoted by x_y) and a greatest lower bound (denoted by x ^ y)), then the p.o. -
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. -
The Theory of Lattice-Ordered Groups
The Theory ofLattice-Ordered Groups Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 307 The Theory of Lattice-Ordered Groups by V. M. Kopytov Institute ofMathematics, RussianAcademyof Sciences, Siberian Branch, Novosibirsk, Russia and N. Ya. Medvedev Altai State University, Bamaul, Russia Springer-Science+Business Media, B.Y A C.I.P. Catalogue record for this book is available from the Library ofCongress. ISBN 978-90-481-4474-7 ISBN 978-94-015-8304-6 (eBook) DOI 10.1007/978-94-015-8304-6 Printed on acid-free paper All Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994. Softcover reprint ofthe hardcover Ist edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrie val system, without written permission from the copyright owner. Contents Preface IX Symbol Index Xlll 1 Lattices 1 1.1 Partially ordered sets 1 1.2 Lattices .. ..... 3 1.3 Properties of lattices 5 1.4 Distributive and modular lattices. Boolean algebras 6 2 Lattice-ordered groups 11 2.1 Definition of the l-group 11 2.2 Calculations in I-groups 15 2.3 Basic facts . 22 3 Convex I-subgroups 31 3.1 The lattice of convex l-subgroups .......... .. 31 3.2 Archimedean o-groups. Convex subgroups in o-groups. 34 3.3 Prime subgroups 39 3.4 Polars ... ..................... 43 3.5 Lattice-ordered groups with finite Boolean algebra of polars ...................... -
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 . -
Groups of Divisibility
University of South Dakota USD RED Honors Thesis Theses, Dissertations, and Student Projects Spring 3-5-2020 Groups of Divisibility Seth J. Gerberding University of South Dakota Follow this and additional works at: https://red.library.usd.edu/honors-thesis Part of the Algebra Commons Recommended Citation Gerberding, Seth J., "Groups of Divisibility" (2020). Honors Thesis. 76. https://red.library.usd.edu/honors-thesis/76 This Honors Thesis is brought to you for free and open access by the Theses, Dissertations, and Student Projects at USD RED. It has been accepted for inclusion in Honors Thesis by an authorized administrator of USD RED. For more information, please contact [email protected]. Groups of Divisibility by Seth Gerberding A thesis submitted in partial fulfillment of the requirements for the University Honors Program Department of Mathematical Sciences The University of South Dakota May 2020 The members of the Honors Thesis Committee appointed to examine the thesis of Seth Gerberding find it satisfactory and recommend that it be accepted. Dr. Ramiro Lafuente-Rodriguez Department of Mathematical Sciences Director of the Committee Dr. Gabriel Picioroaga Department of Mathematical Sciences Dr. Dan Van Peursem Department of Mathematical Sciences ii ABSTRACT Groups of Divisibility Seth Gerberding Director: Ramiro Lafuente-Rodriguez, Ph.D. In this thesis, we examine a part of abstract algebra known as Groups of Divisibility. We construct these special groups from basic concepts. We begin with partially-ordered sets, then build our way into groups, rings, and even structures akin to rings of poly- nomials. In particular, we explore how elementary algebra evolves when an ordering is included with the operations. -
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 . -
Studying Monoids Is Not Enough to Study Multiplicative Properties of Rings: an Elementary Approach
Arab. J. Math. (2015) 4:29–34 DOI 10.1007/s40065-014-0118-1 Arabian Journal of Mathematics Marco Fontana · Muhammad Zafrullah Studying monoids is not enough to study multiplicative properties of rings: an elementary approach Received: 15 April 2014 / Accepted: 27 August 2014 / Published online: 25 September 2014 © The Author(s) 2014. This article is published with open access at Springerlink.com Abstract The aim of these notes is to indicate, using very simple examples, that not all results in ring theory can be derived from monoids and that there are results that deeply depend on the interplay between “+”and “·”. Mathematics Subject Classification 20M14 · 20M12 · 20M25 · 13A15 · 13G05 In a commutative unitary ring (R, +, ·), with 0 = 1, there are two binary operations, denoted by “+”, addition, and by “·”, multiplication, at work. So, in particular, we can consider R as an algebraic system closed under one of the binary operations, that is (R, +) or (R, ·). In both cases, R is at least a monoid. This state of affairs could lead one to think that perhaps it is sufficient to study monoids to get a handle on ring theory. The aim of these notes is to indicate, using elementary and easily accessible results available in literature, that not all results in ring theory can be derived separately from semigroups or monoids and that there are results that deeply depend on the interplay between “+”and“·”. Following the custom, we choose to use the simplest examples that can be developed with a minimum of jargon to establish our thesis. We shall restrict our attention to integral domains and to results of multiplicative nature, as that is our area of interest. -
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. -
Chapter 8 Ordered Sets
Chapter VIII Ordered Sets, Ordinals and Transfinite Methods 1. Introduction In this chapter, we will look at certain kinds of ordered sets. If a set \ is ordered in a reasonable way, then there is a natural way to define an “order topology” on \. Most interesting (for our purposes) will be ordered sets that satisfy a very strong ordering condition: that every nonempty subset contains a smallest element. Such sets are called well-ordered. The most familiar example of a well-ordered set is and it is the well-ordering property that lets us do mathematical induction in In this chapter we will see “longer” well ordered sets and these will give us a new proof method called “transfinite induction.” But we begin with something simpler. 2. Partially Ordered Sets Recall that a relation V\ on a set is a subset of \‚\ (see Definition I.5.2 ). If ÐBßCÑ−V, we write BVCÞ An “order” on a set \ is refers to a relation on \ that satisfies some additional conditions. Order relations are usually denoted by symbols such asŸ¡ß£ , , or . Definition 2.1 A relation V\ on is called: transitive ifÀ a +ß ,ß - − \ Ð+V, and ,V-Ñ Ê +V-Þ reflexive ifÀa+−\+V+ antisymmetric ifÀ a +ß , − \ Ð+V, and ,V+ Ñ Ê Ð+ œ ,Ñ symmetric ifÀ a +ß , − \ +V, Í ,V+ (that is, the set V is “symmetric” with respect to thediagonal ? œÖÐBßBÑÀB−\ש\‚\). Example 2.2 1) The relation “œ\ ” on a set is transitive, reflexive, symmetric, and antisymmetric. Viewed as a subset of \‚\, the relation “ œ ” is the diagonal set ? œÖÐBßBÑÀB−\×Þ 2) In ‘, the usual order relation is transitive and antisymmetric, but not reflexive or symmetric.