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Relations in Categories
Relations in Categories Stefan Milius A thesis submitted to the Faculty of Graduate Studies in partial fulfilment of the requirements for the degree of Master of Arts Graduate Program in Mathematics and Statistics York University Toronto, Ontario June 15, 2000 Abstract This thesis investigates relations over a category C relative to an (E; M)-factori- zation system of C. In order to establish the 2-category Rel(C) of relations over C in the first part we discuss sufficient conditions for the associativity of horizontal composition of relations, and we investigate special classes of morphisms in Rel(C). Attention is particularly devoted to the notion of mapping as defined by Lawvere. We give a significantly simplified proof for the main result of Pavlovi´c,namely that C Map(Rel(C)) if and only if E RegEpi(C). This part also contains a proof' that the category Map(Rel(C))⊆ is finitely complete, and we present the results obtained by Kelly, some of them generalized, i. e., without the restrictive assumption that M Mono(C). The next part deals with factorization⊆ systems in Rel(C). The fact that each set-relation has a canonical image factorization is generalized and shown to yield an (E¯; M¯ )-factorization system in Rel(C) in case M Mono(C). The setting without this condition is studied, as well. We propose a⊆ weaker notion of factorization system for a 2-category, where the commutativity in the universal property of an (E; M)-factorization system is replaced by coherent 2-cells. In the last part certain limits and colimits in Rel(C) are investigated. -
Simple Laws About Nonprominent Properties of Binary Relations
Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt jochen.burghardt alumni.tu-berlin.de Nov 2018 Abstract We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like \not irreflexive, but asymmetric". We considered the laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties. Keywords: Binary relation; Quine-McCluskey algorithm; Hypotheses generation arXiv:1806.05036v2 [math.LO] 20 Nov 2018 Contents 1 Introduction 4 2 Definitions 8 3 Reported law suggestions 10 4 Formal proofs of property laws 21 4.1 Co-reflexivity . 21 4.2 Reflexivity . 23 4.3 Irreflexivity . 24 4.4 Asymmetry . 24 4.5 Symmetry . 25 4.6 Quasi-transitivity . 26 4.7 Anti-transitivity . 28 4.8 Incomparability-transitivity . 28 4.9 Euclideanness . 33 4.10 Density . 38 4.11 Connex and semi-connex relations . 39 4.12 Seriality . 40 4.13 Uniqueness . 42 4.14 Semi-order property 1 . 43 4.15 Semi-order property 2 . 45 5 Examples 48 6 Implementation issues 62 6.1 Improved relation enumeration . 62 6.2 Quine-McCluskey implementation . 64 6.3 On finding \nice" laws . 66 7 References 69 List of Figures 1 Source code for transitivity check . .5 2 Source code to search for right Euclidean non-transitive relations . .5 3 Timing vs. universe cardinality . -
What Are Kinship Terminologies, and Why Do We Care?: a Computational Approach To
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Kent Academic Repository What are Kinship Terminologies, and Why do we Care?: A Computational Approach to Analysing Symbolic Domains Dwight Read, UCLA Murray Leaf, University of Texas, Dallas Michael Fischer, University of Kent, Canterbury, Corresponding Author, [email protected] Abstract Kinship is a fundamental feature and basis of human societies. We describe a set of computat ional tools and services, the Kinship Algebra Modeler, and the logic that underlies these. Thes e were developed to improve how we understand both the fundamental facts of kinship, and h ow people use kinship as a resource in their lives. Mathematical formalism applied to cultural concepts is more than an exercise in model building, as it provides a way to represent and exp lore logical consistency and implications. The logic underlying kinship is explored here thro ugh the kin term computations made by users of a terminology when computing the kinship r elation one person has to another by referring to a third person for whom each has a kin term relationship. Kinship Algebra Modeler provides a set of tools, services and an architecture to explore kinship terminologies and their properties in an accessible manner. Keywords Kinship, Algebra, Semantic Domains, Kinship Terminology, Theory Building 1 1. Introduction The Kinship Algebra Modeller (KAM) is a suite of open source software tools and services u nder development to support the elicitation and analysis of kinship terminologies, building al gebraic models of the relations structuring terminologies, and instantiating these models in lar ger contexts to better understand how people pragmatically interpret and employ the logic of kinship relations as a resource in their individual and collective lives. -
A General Account of Coinduction Up-To Filippo Bonchi, Daniela Petrişan, Damien Pous, Jurriaan Rot
A General Account of Coinduction Up-To Filippo Bonchi, Daniela Petrişan, Damien Pous, Jurriaan Rot To cite this version: Filippo Bonchi, Daniela Petrişan, Damien Pous, Jurriaan Rot. A General Account of Coinduction Up-To. Acta Informatica, Springer Verlag, 2016, 10.1007/s00236-016-0271-4. hal-01442724 HAL Id: hal-01442724 https://hal.archives-ouvertes.fr/hal-01442724 Submitted on 20 Jan 2017 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. A General Account of Coinduction Up-To ∗ Filippo Bonchi Daniela Petrişan Damien Pous Jurriaan Rot May 2016 Abstract Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modeled as coalgebras. The fact that bisimulations up to context can be safely used in any language specified by GSOS rules can also be seen as an instance of our framework, using the well-known observation by Turi and Plotkin that such languages form bialgebras. -
Foundations of Relations and Kleene Algebra
Introduction Foundations of Relations and Kleene Algebra Aim: cover the basics about relations and Kleene algebras within the framework of universal algebra Peter Jipsen This is a tutorial Slides give precise definitions, lots of statements Decide which statements are true (can be improved) Chapman University which are false (and perhaps how they can be fixed) [Hint: a list of pages with false statements is at the end] September 4, 2006 Peter Jipsen (Chapman University) Relation algebras and Kleene algebra September 4, 2006 1 / 84 Peter Jipsen (Chapman University) Relation algebras and Kleene algebra September 4, 2006 2 / 84 Prerequisites Algebraic properties of set operation Knowledge of sets, union, intersection, complementation Some basic first-order logic Let U be a set, and P(U)= {X : X ⊆ U} the powerset of U Basic discrete math (e.g. function notation) P(U) is an algebra with operations union ∪, intersection ∩, These notes take an algebraic perspective complementation X − = U \ X Satisfies many identities: e.g. X ∪ Y = Y ∪ X for all X , Y ∈ P(U) Conventions: How can we describe the set of all identities that hold? Minimize distinction between concrete and abstract notation x, y, z, x1,... variables (implicitly universally quantified) Can we decide if a particular identity holds in all powerset algebras? X , Y , Z, X1,... set variables (implicitly universally quantified) These are questions about the equational theory of these algebras f , g, h, f1,... function variables We will consider similar questions about several other types of algebras, a, b, c, a1,... constants in particular relation algebras and Kleene algebras i, j, k, i1,.. -
Anti-Conformism and Social Networks Alexis Poindron
Anti-conformism and Social Networks Alexis Poindron To cite this version: Alexis Poindron. Anti-conformism and Social Networks. Economics and Finance. 2016. dumas- 02102411 HAL Id: dumas-02102411 https://dumas.ccsd.cnrs.fr/dumas-02102411 Submitted on 18 Apr 2019 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. Anti-conformism and Social Networks Alexis Poindron Université Paris I Sorbonne, UFR de sciences économiques 02 Master 2 Économie Théorique et Empirique 2015-2016 Mémoire présenté et soutenu par Alexis Poindron le 17/05/2016 Sous la direction de Michel Grabisch et Agnieszka Rusinowska May 17, 2016 L’Université de paris I Panthéon Sorbonne n’entend donner aucune approbation, ni désappro- bation aux opinions émises dans ce mémoire ; elle doivent être considérées comme propres à leur auteur. 1 Contents 1 Introduction . .3 1.1 Literature review and context of this master thesis . .3 1.2 Some notations and definitions . .5 2 Generalized OWA . .6 2.1 GOWA: definitions and exemples . .6 2.2 Discussion on GOWA . .8 3 Terminal states and classes with GOWA: homogeneous framework . .9 3.1 Preliminary results . .9 3.2 List of possible terminal classes and unicity . -
Introduction to Relations
CHAPTER 7 Introduction to Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition:A binary relation from a set A to a set B is a subset R ⊆ A × B: If (a; b) 2 R we say a is related to b by R. A is the domain of R, and B is the codomain of R. If A = B, R is called a binary relation on the set A. Notation: • If (a; b) 2 R, then we write aRb. • If (a; b) 62 R, then we write aR6 b. Discussion Notice that a relation is simply a subset of A × B. If (a; b) 2 R, where R is some relation from A to B, we think of a as being assigned to b. In these senses students often associate relations with functions. In fact, a function is a special case of a relation as you will see in Example 1.2.4. Be warned, however, that a relation may differ from a function in two possible ways. If R is an arbitrary relation from A to B, then • it is possible to have both (a; b) 2 R and (a; b0) 2 R, where b0 6= b; that is, an element in A could be related to any number of elements of B; or • it is possible to have some element a in A that is not related to any element in B at all. 204 1. RELATIONS AND THEIR PROPERTIES 205 Often the relations in our examples do have special properties, but be careful not to assume that a given relation must have any of these properties. -
Social Networks IZA Dpno
IZA DP No. 4621 Social Networks Joan de Martí Yves Zenou December 2009 DISCUSSION PAPER SERIES Forschungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor Social Networks Joan de Martí Universitat Pompeu Fabra and Barcelona GSE Yves Zenou Stockholm University, IFN and IZA Discussion Paper No. 4621 December 2009 IZA P.O. Box 7240 53072 Bonn Germany Phone: +49-228-3894-0 Fax: +49-228-3894-180 E-mail: [email protected] Any opinions expressed here are those of the author(s) and not those of IZA. Research published in this series may include views on policy, but the institute itself takes no institutional policy positions. The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center and a place of communication between science, politics and business. IZA is an independent nonprofit organization supported by Deutsche Post Foundation. The center is associated with the University of Bonn and offers a stimulating research environment through its international network, workshops and conferences, data service, project support, research visits and doctoral program. IZA engages in (i) original and internationally competitive research in all fields of labor economics, (ii) development of policy concepts, and (iii) dissemination of research results and concepts to the interested public. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author. IZA Discussion Paper No. 4621 December 2009 ABSTRACT Social Networks We survey the literature on social networks by putting together the economics, sociological and physics/applied mathematics approaches, showing their similarities and differences. -
The Algebraic Theory of Semigroups the Algebraic Theory of Semigroups
http://dx.doi.org/10.1090/surv/007.1 The Algebraic Theory of Semigroups The Algebraic Theory of Semigroups A. H. Clifford G. B. Preston American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 20-XX. International Standard Serial Number 0076-5376 International Standard Book Number 0-8218-0271-2 Library of Congress Catalog Card Number: 61-15686 Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © Copyright 1961 by the American Mathematical Society. All rights reserved. Printed in the United States of America. Second Edition, 1964 Reprinted with corrections, 1977. The American Mathematical Society retains all rights except those granted to the United States Government. -
The Mathematical Sociologist
The Mathematical Sociologist Newsletter of the Mathematical Sociology Section of the American Sociological Association Spring 2003 Section Officers Chair From the newsletter editor Noah E. Friedkin, Barbara Meeker, [email protected] University of California, Santa Barbara The MathSoc Section Newsletter is [email protected] back by popular demand (sort of). After a gap of Chair Elect: about two years, I volunteered once again to David Heise, Indiana University produce the newsletter. Given the many very [email protected] competitive professional accomplishments we all Past Chair, and: Section Nominations try for, it was something of a relief to campaign Committee Chair for a job absolutely no one else wants! I intend to Patrick Doreian, University of Pittsburgh do either one or two issues a year. This is the [email protected] issue for 2002-2003, covering approximately Secretary/Treasurer – 2005 April 2002 through April 2003. Dave McFarland Lisa Troyer, University of Iowa (UCLA - [email protected]) [email protected] has agreed to post newsletters and other information on a Section web page. Council 2003 The Section has been very active, as Kenneth C. Land, Duke University shown in the minutes of the Business Meeting, [email protected] prepared by 2002 Secretary-Treasurer Joe Whitmeyer, (see below). Other activities of the Jane Sell, Texas A&M University Section during the past year and some with ASA [email protected] coming up are also described below. Section Chair Noah Friedkin has written Council - 2004 an interesting comment on the Current State of Diane H. Felmlee, Mathematical Sociology; I think most of us University of California-Davis would agree wholeheartedly with his remarks. -
From Peirce's Algebra of Relations to Tarski's Calculus of Relations
The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations Author(s): Roger D. Maddux Source: Studia Logica: An International Journal for Symbolic Logic, Vol. 50, No. 3/4, Algebraic Logic (1991), pp. 421-455 Published by: Springer Stable URL: http://www.jstor.org/stable/20015596 Accessed: 02/03/2009 14:54 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=springer. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. Springer is collaborating with JSTOR to digitize, preserve and extend access to Studia Logica: An International Journal for Symbolic Logic. -
Regular Entailment Relations
Regular entailment relations Introduction If G is an ordered commutative group and we have a map f : G ! L where L is a l-group, we can define a relation A ` B between non empty finite sets of G by ^f(A) 6 _f(B). This relation satisfies the conditions 1. a ` b if a 6 b in G 2. A ` B if A ⊇ A0 and B ⊇ B0 and A0 ` B0 3. A ` B if A; x ` B and A ` B; x 4. A ` B if A + x ` B + x 5. a + x; b + y ` a + b; x + y We call a regular entailment relation on an ordered group G any relation which satisfies these condi- tions. The remarkable last condition is called the regularity condition. Note that the converse relation of a regular entailment relation is a regular entailment relation. Any relation satisfying the three first conditions define in a canonical way a (non bounded) distributive lattice L. The goal of this note is to show that this distributive lattice has a (canonical) l-group structure. 1 General properties A first consequence of regularity is the following. Proposition 1.1 We have a; b ` a + x; b − x and a + x; b − x ` a; b. In particular, a ` a + x; a − x and a + x; a − x ` a Proof. By regularity we have (−x + a + x); (b + 2x − 2x) ` (−x + b + 2x); (a + x − 2x). The other claim is symmetric. Corollary 1.2 ^A 6 (^A + x) _ (^A − x). Proof. We can reason in the distributive lattice L defined by the given (non bounded) entailment relation and use Proposition 7.3.