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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. -
A Survey of Abstract Algebraic Logic 15 Bras
J. M. Font A Survey of Abstract Algebraic R. Jansana D. Pigozzi Logic Contents Introduction 14 1. The First Steps 16 1.1. Consequence operations and logics . 17 1.2. Logical matrices . 20 1.3. Lindenbaum-Tarski algebras . 22 2. The Lindenbaum-Tarski Process Fully Generalized 24 2.1. Frege's principles . 24 2.2. Algebras and matrices canonically associated with a logic . 26 3. The Core Theory of Abstract Algebraic Logic 29 3.1. Elements of the general theory of matrices . 29 3.2. Protoalgebraic logics . 33 3.3. Algebraizable logics . 38 3.4. The Leibniz hierarchy . 43 4. Extensions of the Core Theory 50 4.1. k-deductive systems and universal algebra . 50 4.2. Gentzen systems and their generalizations . 55 4.3. Equality-free universal Horn logic . 61 4.4. Algebras of Logic . 64 5. Generalized Matrix Semantics and Full Models 71 6. Extensions to More Complex Notions of Logic 79 6.1. The semantics-based approach . 80 6.2. The categorial approach . 85 References 88 Special Issue on Abstract Algebraic Logic { Part II Edited by Josep Maria Font, Ramon Jansana, and Don Pigozzi Studia Logica 74: 13{97, 2003. c 2003 Kluwer Academic Publishers. Printed in the Netherlands. 14 J. M. Font, R. Jansana, and D. Pigozzi Introduction Algebraic logic was born in the XIXth century with the work of Boole, De Morgan, Peirce, Schr¨oder, etc. on classical logic, see [12, 16]. They took logical equivalence rather than truth as the primitive logical predicate, and, exploiting the similarity between logical equivalence and equality, they devel- oped logical systems in which metalogical investigations take on a distinctly algebraic character. -
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. -
What Is Abstract Algebraic Logic?
What is Abstract Algebraic Logic? Umberto Rivieccio University of Genoa Department of Philosophy Via Balbi 4 - Genoa, Italy email [email protected] Abstract Our aim is to answer the question of the title, explaining which kind of problems form the nucleus of the branch of algebraic logic that has come to be known as Abstract Algebraic Logic. We give a concise account of the birth and evolution of the field, that led from the discovery of the connections between certain logical systems and certain algebraic structures to a general theory of algebraization of logics. We consider first the so-called algebraizable logics, which may be regarded as a paradigmatic case in Abstract Algebraic Logic; then we shift our attention to some wider classes of logics and to the more general framework in which they are studied. Finally, we review some other topics in Abstract Algebraic Logic that, though equally interesting and promising for future research, could not be treated here in detail. 1 Introduction Algebraic logic can be described in very general terms as the study of the relations between algebra and logic. One of the main reasons that motivate such a study is the possibility to treat logical problems with algebraic methods. This means that, in order to solve a logical problem, we may try to translate it into algebraic terms, find the solution using algebraic techniques, and then translate back the result into logic. This strategy proved to be fruitful, since algebra is an old and well{established field of mathematics, that offers powerful tools to treat problems that would possibly be much harder to solve using the logical machinery alone. -
Universal Algebra
Donald Sannella and Andrzej Tarlecki Foundations of Algebraic Specification and Formal Software Development September 29, 2010 Springer Page: v job: root macro: svmono.cls date/time: 29-Sep-2010/17:45 Page: xiv job: root macro: svmono.cls date/time: 29-Sep-2010/17:45 Contents 0 Introduction .................................................... 1 0.1 Modelling software systems as algebras . 1 0.2 Specifications . 5 0.3 Software development . 8 0.4 Generality and abstraction . 10 0.5 Formality . 12 0.6 Outlook . 14 1 Universal algebra .............................................. 15 1.1 Many-sorted sets . 15 1.2 Signatures and algebras . 18 1.3 Homomorphisms and congruences . 22 1.4 Term algebras . 27 1.5 Changing signatures . 32 1.5.1 Signature morphisms . 32 1.5.2 Derived signature morphisms . 36 1.6 Bibliographical remarks . 38 2 Simple equational specifications ................................. 41 2.1 Equations . 41 2.2 Flat specifications . 44 2.3 Theories . 50 2.4 Equational calculus . 54 2.5 Initial models . 58 2.6 Term rewriting . 66 2.7 Fiddling with the definitions . 72 2.7.1 Conditional equations . 72 2.7.2 Reachable semantics . 74 2.7.3 Dealing with partial functions: error algebras . 78 2.7.4 Dealing with partial functions: partial algebras . 84 2.7.5 Partial functions: order-sorted algebras . 87 xv xvi Contents 2.7.6 Other options . 91 2.8 Bibliographical remarks . 93 3 Category theory................................................ 97 3.1 Introducing categories. 99 3.1.1 Categories . 99 3.1.2 Constructing categories . 105 3.1.3 Category-theoretic definitions . 109 3.2 Limits and colimits . 111 3.2.1 Initial and terminal objects . -
What Is a Logic Translation? in Memoriam Joseph Goguen
, 1{29 c 2009 Birkh¨auserVerlag Basel/Switzerland What is a Logic Translation? In memoriam Joseph Goguen Till Mossakowski, R˘azvan Diaconescu and Andrzej Tarlecki Abstract. We study logic translations from an abstract perspective, with- out any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof-theoretic and model- theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quanti- fiers, although not present in the definition of logic and logic translation, can be recovered by their abstract properties and are preserved and reflected by translations under suitable conditions. 1. Introduction The development of a notion of logic translation presupposes the development of a notion of logic. Logic has been characterised as the study of sound reasoning, of what follows from what. Hence, the notion of logical consequence is central. It can be obtained in two complementary ways: via the model-theoretic notion of satisfaction in a model and via the proof-theoretic notion of derivation according to proof rules. These notions can be captured in a completely abstract manner, avoiding any particular commitment to the nature of models, sentences, or rules. The paper [40], which is the predecessor of our current work, explains the notion of logic along these lines. Similarly, an abstract notion of logic translation can be developed, both at the model-theoretic and at the proof-theoretic level. A central question concern- ing such translations is their interaction with logical structure, such as given by logical connectives, and logical properties. -
Duality Theory and Abstract Algebraic Logic
Duality Theory and Abstract Algebraic Logic María Esteban ADVERTIMENT. La consulta d’aquesta tesi queda condicionada a l’acceptació de les següents condicions d'ús: La difusió d’aquesta tesi per mitjà del servei TDX (www.tdx.cat) i a través del Dipòsit Digital de la UB (diposit.ub.edu) ha estat autoritzada pels titulars dels drets de propietat intel·lectual únicament per a usos privats emmarcats en activitats d’investigació i docència. No s’autoritza la seva reproducció amb finalitats de lucre ni la seva difusió i posada a disposició des d’un lloc aliè al servei TDX ni al Dipòsit Digital de la UB. No s’autoritza la presentació del seu contingut en una finestra o marc aliè a TDX o al Dipòsit Digital de la UB (framing). Aquesta reserva de drets afecta tant al resum de presentació de la tesi com als seus continguts. En la utilització o cita de parts de la tesi és obligat indicar el nom de la persona autora. ADVERTENCIA. La consulta de esta tesis queda condicionada a la aceptación de las siguientes condiciones de uso: La difusión de esta tesis por medio del servicio TDR (www.tdx.cat) y a través del Repositorio Digital de la UB (diposit.ub.edu) ha sido autorizada por los titulares de los derechos de propiedad intelectual únicamente para usos privados enmarcados en actividades de investigación y docencia. No se autoriza su reproducción con finalidades de lucro ni su difusión y puesta a disposición desde un sitio ajeno al servicio TDR o al Repositorio Digital de la UB. -
On Birkhoff's Common Abstraction Problem
F. Paoli On Birkho®'s Common C. Tsinakis Abstraction Problem Abstract. In his milestone textbook Lattice Theory, Garrett Birkho® challenged his readers to develop a \common abstraction" that includes Boolean algebras and lattice- ordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of BA and LG their join BA_LG in the lattice of subvarieties of FL (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for BA_LG relative to FL. Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing BA _ LG. Keywords: Residuated lattice, FL-algebra, Substructural logics, Boolean algebra, Lattice- ordered group, Birkho®'s problem, History of 20th C. algebra. 1. Introduction In his milestone textbook Lattice Theory [2, Problem 108], Garrett Birkho® challenged his readers by suggesting the following project: Develop a common abstraction that includes Boolean algebras (rings) and lattice ordered groups as special cases. Over the subsequent decades, several mathematicians tried their hands at Birkho®'s intriguing problem. Its very formulation, in fact, intrinsically seems to call for reiterated attempts: unlike most problems contained in the book, for which it is manifest what would count as a correct solution, this one is stated in su±ciently vague terms as to leave it open to debate whether any proposed answer is really adequate. It appears to us that Rama Rao puts things right when he remarks [28, p. -
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. -
How Peircean Was the “'Fregean' Revolution” in Logic?
HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0.