DOCSLIB.ORG

Duality Theory and Abstract Algebraic Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. No se autoriza la presentación de su contenido en una ventana o marco ajeno a TDR o al Repositorio Digital de la UB (framing). Esta reserva de derechos afecta tanto al resumen de presentación de la tesis como a sus contenidos. En la utilización o cita de partes de la tesis es obligado indicar el nombre de la persona autora. WARNING. On having consulted this thesis you’re accepting the following use conditions: Spreading this thesis by the TDX (www.tdx.cat) service and by the UB Digital Repository (diposit.ub.edu) has been authorized by the titular of the intellectual property rights only for private uses placed in investigation and teaching activities. Reproduction with lucrative aims is not authorized nor its spreading and availability from a site foreign to the TDX service or to the UB Digital Repository. Introducing its content in a window or frame foreign to the TDX service or to the UB Digital Repository is not authorized (framing). Those rights affect to the presentation summary of the thesis as well as to its contents. In the using or citation of parts of the thesis it’s obliged to indicate the name of the author. DUALITY THEORY AND ABSTRACT ALGEBRAIC LOGIC Mar´ıa Esteban PhD program: Pure and Applied Logic Supervisors: Prof. dr. Ramon Jansana and Prof. dr. Sergio A. Celani Department of Logic, History and Philosophy of Science Faculty of Philosophy Mar´ıaEsteban: Duality theory and Abstract Algebraic Logic, September 2013 supervisors: Prof. dr. Ramon Jansana Prof. dr. Sergio A. Celani committee members: Prof. dr. Josep Maria Font i Llovet Prof. dr. Mai Gehrke Prof. dr. Alessandra Palmigiano This research was supported by the Spanish Ministry of Education, Culture and Sport (FPU Program) grant AP2008-03838 A mis padres y a mis hermanos A Juan Abstract In this thesis we present the results of our research on duality theory for non- classical logics under the point of view of Abstract Algebraic Logic. Firstly, we propose an abstract Spectral-like duality and an abstract Priestley-style duality for every filter distributive finitary congruential logic with theorems. This proposal aims to unify the various dualities for concrete logics that we find in the literature, by showing the abstract template in which all of them fit. Secondly, the dual correspondence of some logical properties is examined. This serves to reveal the connection between our abstract dualities and the concrete dualities related to concrete logics. We apply those results to get new dualities for suitable expansions of a well-known logic: the implicative fragment of intuitionistic logic. Finally, we develop a new technique that can be modularly applied to simplify some of the obtained dualities. Resumen En esta tesis presentamos los resultados de nuestra investigaci´onacerca de la teor´ıade la dualidad para l´ogicasno cl´asicasdesde el punto de vista de la L´ogica Algebr´aicaAbstracta. En primer lugar, proponemos una dualidad abstracta de tipo espectral y otra dualidad abstracta de tipo Priestley para cada l´ogicacongru- encial, filtro distributiva, finitaria y con teoremas. Esta propuesta pretende unificar las distintas dualidades para l´ogicasno cl´asicasque encontramos en la literatura, mostrando el esquema abstracto en el que todas ellas encajan. En segundo lugar, la correspondencia dual de algunas propiedades l´ogicases examinada. Esto sirve para revelar la conexi´onque existe entre nuestras dualidades abstractas y las du- alidades concretas relacionadas con l´ogicasconcretas. Aplicamos estos resultados para obtener nuevas dualidades para expansiones apropiadas de una l´ogicabien conocida: el fragmento implicativo de la l´ogicaintuicionista. Finalmente, desarrol- lamos una nueva t´ecnicaque puede ser aplicada de forma modular para simplificar algunas de las dualidades obtenidas. v Acknowledgments I would like to express my deepest gratitude to my supervisor, Ramon Jansana, for his excellent guidance, caring, patience, and providing me with an excellent atmosphere for doing research, not to mention his unsurpassed knowledge of duality theory and Abstract Algebraic Logic. The good advice, support and friendship of my second supervisor, Sergio A. Celani, has been invaluable on both an academic and a personal level, for which I am extremely grateful. I would also like to express my gratitude to the members of my thesis com- mittee: Josep Mar´ıaFont, Mai Gehrke and Alessandra Palmigiano. Thank you very much for having accepted the task of critically reading this thesis. I have also profited from discussion with the members of the Barcelona research group on Non-classical logics, and with audiences in Oxford, Sevilla, Nashville and Orange. Thanks to a Master fellowship from Obra Social Caja Espa~na,I could come to study the Master in Barcelona, and thanks to a FPU doctoral grant from the spanish Ministry of Education, Culture and Sport, I could stay in Barcelona four more years and finish my PhD. During these years I met a lot of people along the way, that made that the task that lay ahead was more enjoyable. This dissertation could not have been written without the support and friendship found at Barcelona. Pablo welcomed me so charming at the very beginning and Marta and Mar´ıahelped me a lot at the very end. Nuria and Manel provided me with a home, and it was so nice to have Bernat and Nuno as neighbors. I will miss so much lunch at el office with Luz, Pepe, Viviane, Blanca, Sergi, Luciano, Daniel, Gon¸calo,...I would like to thank you all for being there, and also the people of the Logic Team and of Logos. My research has gotten a great benefit from the travels I have undertaken. Apart from the conferences I attended, during my PhD I have made three stays abroad that have been very profitable from both an academic and a personal per- spective. The first one was at the ILLC in Amsterdam during fall 2010, under the supervision of the professor Alessandra Palmigiano. The second one was in Tandil during fall 2011, under the supervision of my co-supervisor Sergio A. Celani. The third one was in Paris during winter 2013, under the supervision of the professor Mai Gehrke. I feel very lucky to have had the opportunity of working in such diffe- rent environments (philosophy, mathematics and computer science). Any student who has the opportunity should take advantage of it. In Amsterdam (and also later on in Barcelona) I had the opportunity of dis- cussing with Alessandra, whose work has been very inspiring to this dissertation. I could also feel the great atmosphere of the ILLC, and I met many people that made the stay so great. I would like to thank all of them, and specially Margot. In Tandil (and also later on in Barcelona) I had the pleasure of working with Sergio, that also shared with me asados and walks. Ramon, Ismael, Felix and Carles helped to make the stay nicer. In Paris I had the chance of discussing and working with Mai, whose enthusiasm and devotion about mathematics makes any conversation with her amazing. I also enjoyed a lot the discussions with Sam, that definitely vii transmits the same passion about what he does. Besides, Margot and my flatmates at T´el´egraphemade so pleasant my stay there. I would like to acknowledge the financial, academic and technical support of the University of Barcelona and its staff, specially Pablo R´ıo,Oscar Cabaco and Paco Murcia, who were always willing to help me. Luckily I could count on the help of logalg, thank you very much to all who make it possible, notably to Felix Bou. I would also like to thank the online LATEX community that has made the actual process of writing and typesetting this dissertation so much easier, by developing the software and helping newbies like me to understand and benefit from them. Por ´ultimo,querr´ıadar las gracias a mi familia y amigos. Especialmente a mi t´ıaVirginia, porque me ayud´oenormemente al leer y revisar algunas partes del manuscrito. Tambi´ena mis padres, Antonio y Visi, porque siempre han cre´ıdoen m´ıy me han ayudado a alcanzar mis metas.
Load more

Recommended publications
  • Topological Duality and Lattice Expansions Part I: a Topological Construction of Canonical Extensions
    TOPOLOGICAL DUALITY AND LATTICE EXPANSIONS PART I: A TOPOLOGICAL CONSTRUCTION OF CANONICAL EXTENSIONS M. ANDREW MOSHIER AND PETER JIPSEN 1. INTRODUCTION The two main objectives of this paper are (a) to prove topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. The paper is first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper. Regarding objective (a), consider the following simple question: Is there a subcategory of Top that is dually equivalent to Lat? Here, Top is the category of topological spaces and continuous maps and Lat is the category of bounded lattices and lattice homomorphisms. To date, the question has been answered positively either by specializing Lat or by generalizing Top. The earliest examples are of the former sort. Tarski [Tar29] (treated in English, e.g., in [BD74]) showed that every complete atomic Boolean lattice is represented by a powerset. Taking some historical license, we can say this result shows that the category of complete atomic Boolean lattices with complete lat- tice homomorphisms is dually equivalent to the category of discrete topological spaces. Birkhoff [Bir37] showed that every finite distributive lattice is represented by the lower sets of a finite partial order. Again, we can now say that this shows that the category of finite distributive lattices is dually equivalent to the category of finite T0 spaces and con- tinuous maps.
    [Show full text]
  • Distributive Envelopes and Topological Duality for Lattices Via Canonical Extensions
    DISTRIBUTIVE ENVELOPES AND TOPOLOGICAL DUALITY FOR LATTICES VIA CANONICAL EXTENSIONS MAI GEHRKE AND SAM VAN GOOL Abstract. We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the stan- dard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. In the course of the paper, we obtain the following results: (1) canoni- cal extensions of bounded lattices are the algebraic versions of the existing dualities for bounded lattices by Urquhart and Hartung, (2) there is a univer- sal construction which associates to an arbitrary lattice two distributive lattice envelopes with an adjoint pair between them, and (3) we give a topological du- ality for bounded lattices which generalizes Priestley duality and which shows precisely which maps between bounded lattices admit functional duals. For the result in (1), we rely on previous work of Gehrke, J´onssonand Harding. For the universal construction in (2), we modify a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of `admis- sibility' to the finitary case. For (3), we use Priestley duality for distributive lattices and our universal characterization of the distributive envelopes. 1. Introduction Topological duality for Boolean algebras [22] and distributive lattices [23] is a useful tool for studying relational semantics for propositional logics. Canonical extensions [16, 17, 11, 10] provide a way of looking at these semantics algebraically. In the absence of a satisfactory topological duality, canonical extensions have been used [3] to treat relational semantics for substructural logics.
    [Show full text]
  • Bitopological Duality for Distributive Lattices and Heyting Algebras
    BITOPOLOGICAL DUALITY FOR DISTRIBUTIVE LATTICES AND HEYTING ALGEBRAS GURAM BEZHANISHVILI, NICK BEZHANISHVILI, DAVID GABELAIA, ALEXANDER KURZ Abstract. We introduce pairwise Stone spaces as a natural bitopological generalization of Stone spaces—the duals of Boolean algebras—and show that they are exactly the bitopolog- ical duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many alge- braic concepts important for the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, co-Heyting algebras, and bi-Heyting algebras, thus providing two new alternatives of Esakia’s duality. 1. Introduction It is widely considered that the beginning of duality theory was Stone’s groundbreaking work in the mid 30ies on the dual equivalence of the category Bool of Boolean algebras and Boolean algebra homomorphism and the category Stone of compact Hausdorff zero- dimensional spaces, which became known as Stone spaces, and continuous functions. In 1937 Stone [28] extended this to the dual equivalence of the category DLat of bounded distributive lattices and bounded lattice homomorphisms and the category Spec of what later became known as spectral spaces and spectral maps. Spectral spaces provide a generalization of Stone 1 spaces. Unlike Stone spaces, spectral spaces are not Hausdorff (not even T1) , and as a result, are more difficult to work with.
    [Show full text]
  • Lattice Duality: the Origin of Probability and Entropy
    , 1 Lattice Duality: The Origin of Probability and Entropy Kevin H. Knuth NASA Ames Research Center, Mail Stop 269-3, Moffett Field CA 94035-1000, USA Abstract Bayesian probability theory is an inference calculus, which originates from a gen- eralization of inclusion on the Boolean lattice of logical assertions to a degree of inclusion represented by a real number. Dual to this lattice is the distributive lat- tice of questions constructed from the ordered set of down-sets of assertions, which forms the foundation of the calculus of inquiry-a generalization of information theory. In this paper we introduce this novel perspective on these spaces in which machine learning is performed and discuss the relationship between these results and several proposed generalizations of information theory in the literature. Key words: probability, entropy, lattice, information theory, Bayesian inference, inquiry PACS: 1 Introduction It has been known for some time that probability theory can be derived as a generalization of Boolean implication to degrees of implication repre- sented by real numbers [11,12]. Straightforward consistency requirements dic- tate the form of the sum and product rules of probability, and Bayes’ thee rem [11,12,47,46,20,34],which forms the basis of the inferential calculus, also known as inductive inference. However, in machine learning applications it is often times more useful to rely on information theory [45] in the design of an algorithm. On the surface, the connection between information theory and probability theory seems clear-information depends on entropy and entropy Email address: kevin.h. [email protected] (Kevin H.
    [Show full text]
  • The Dynamics of Duality: a Fresh Look at the Philosophy of Duality
    数理解析研究所講究録 77 第2050巻 2017年 77-99 The Dynamics of Duality: A Fresh Look at the Philosophy of Duality Yoshihiro Maruyama The Hakubi Centre and Graduate School of Letters, Kyoto University Mathematical, Physical, and Life Sciences Division, University of Oxford [\mathrm{T}]\mathrm{h}\mathrm{e} development of philosophy since the Renaissance has by and large gone from right to left Particularly in physics, this development has reached a peak in our own time, in that, to a large extent, the possibility of knowledge of the objectivisable states of affairs is denied, and it is asserted that we must be content to predict results of observations. This is really the end of all theo‐ retical science in the usual sense (Kurt Gödel, The modern development of the foundations of mathematics in the light of philosophy, lecture never delivered)1 1 Unity, Disunity, and Pluralistic Unity Since the modernist killing of natural philosophy seeking a universal conception of the cosmos as a united whole, our system of knowledge has been optimised for the sake of each particular domain, and has accordingly been massively fragmented and disenchanted (in terms of Webers theory of modernity). Today we lack a unified view of the world, living in the age of disunity surrounded by myriads of uncertainties and contingencies, which invade both science and society as exemplified by different sciences of chance, including quantum theory and artificial intelligence on the basis of statistical machine learning, and by the distinctively after‐modern features of risk society (in terms of the theories of late/reflexive/liquid modermity), respectively (this is not necessarily negative like quite some people respect diversity more than uniformity; contemporary art, for example, gets a lot of inspirations from uncertainties and contingencies out there in nature and society).
    [Show full text]
  • Quantifiers and Duality Luca Reggio
    Quantifiers and duality Luca Reggio To cite this version: Luca Reggio. Quantifiers and duality. Logic in Computer Science [cs.LO]. Université Sorbonne Paris Cité, 2018. English. NNT : 2018USPCC210. tel-02459132 HAL Id: tel-02459132 https://tel.archives-ouvertes.fr/tel-02459132 Submitted on 29 Jan 2020 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. THÈSE DE DOCTORAT DE L’UNIVERSITÉ SORBONNE PARIS CITÉ PRÉPARÉE À L’UNIVERSITÉ PARIS DIDEROT ÉCOLE DOCTORALE DES SCIENCES MATHÉMATIQUES DE PARIS CENTRE ED 386 Quantifiers and duality Par: Dirigée par: Luca REGGIO Mai GEHRKE Thèse de doctorat de Mathématiques Présentée et soutenue publiquement à Paris le 10 septembre 2018 Président du jury: Samson ABRAMSKY, Dept. of Computer Science, University of Oxford Rapporteurs: Mamuka JIBLADZE, Razmadze Mathematical Inst., Tbilisi State University Achim JUNG, School of Computer Science, University of Birmingham Directrice de thése: Mai GEHRKE, Lab. J. A. Dieudonné, CNRS et Université Côte d’Azur Co-encadrants de thése: Samuel J. VAN GOOL, ILLC, University of Amsterdam Daniela PETRI ¸SAN, IRIF, Université Paris Diderot Membres invités: Clemens BERGER, Lab. J. A. Dieudonné, Université Côte d’Azur Paul-André MELLIÈS, IRIF, CNRS et Université Paris Diderot ii Titre: Quantificateurs et dualité Résumé: Le thème central de la présente thèse est le contenu sémantique des quantificateurs logiques.
    [Show full text]
  • Lattice Theory Lecture 3 Duality
    Lattice Theory Lecture 3 Duality John Harding New Mexico State University www.math.nmsu.edu/∼JohnHarding.html [email protected] Toulouse, July 2017 Introduction Today we consider alternative ways to view distributive lattices and Boolean algebras. These are examples of what are known as dualities. Dualities in category theory relate one type of mathematical structure to another. They can be key to forming bridges between areas as different as algebra and topology. Stone duality is a primary example. It predates category theory, and is a terrific guide for further study of other dualities. 2 / 44 Various Dualities in Lattice Theory Birkhoff Duality: finite distributive lattices ↔ finite posets Stone Duality: Boolean algebras ↔ certain topological spaces Priestley Duality: distributive lattices ↔ certain ordered top spaces Esakia Duality: Heyting algebras ↔ certain ordered top spaces We consider the first two. In both cases, prime ideals provide our key tool. 3 / 44 Prime Ideals Recap ... Definition For D a distributive lattice, P ⊆ D is a prime ideal if 1. x ≤ y and y ∈ P ⇒ x ∈ P 2. x; y ∈ P ⇒ x ∨ y ∈ P 3. x ∧ y ∈ P ⇒ x ∈ P or y ∈ P or both. P is called trivial if it is equal to either ∅ or all of D. P 4 / 44 Prime Ideals for Finite Distributive Lattices Proposition In a finite lattice L, the non-empty ideals of L are exactly the principal ideals ↓a = {x ∶ x ≤ a} where a ∈ L. Pf Since ideals are closed under binary joins, if I is a non-empty ideal in a finite lattice, the join of all of its elements again lies in I , and is clearly the largest element of I .
    [Show full text]
  • Stone Duality for Markov Processes
    Stone Duality for Markov Processes Dexter Kozen1 Kim G. Larsen2 Radu Mardare2 Prakash Panangaden3 1Computer Science Department, Cornell University, Ithaca, New York, USA 2Department of Computer Science, University of Aalborg, Aalborg, Denmark 3School of Computer Science, McGill University, Montreal, Canada March 14, 2013 Abstract We define Aumann algebras, an algebraic analog of probabilistic modal logic. An Aumann algebra consists of a Boolean algebra with operators modeling probabilistic transitions. We prove that countable Aumann algebras and countably-generated continuous-space Markov processes are dual in the sense of Stone. Our results subsume exist- ing results on completeness of probabilistic modal logics for Markov processes. 1 Introduction The Stone representation theorem [1] is a recognized landmarks of mathe- matics. The theorem states that every (abstract) Boolean algebra is isomor- phic to a Boolean algebra of sets. In fact, Stone proved much more: that Boolean algebras are dual to a certain category of topological spaces known as Stone spaces. See Johnstone [2] for a detailed and eloquent introduction to this theorem and its ramifications, as well as an account of several other related dualities, like Gelfand duality, that appear in mathematics. Formally, Stone duality states that the category of Boolean algebras and Boolean algebra homomorphisms and the category of Stone spaces and continuous maps are contravariantly equivalent. Jonsson and Tarski [3] ex- tended this to modal logic and showed a duality between Boolean algebras 1 with additional operators and certain topological spaces with additional operators. Stone duality embodies completeness theorems, but goes far beyond them. Proofs of completeness typically work by constructing an instance of a model from maximal consistent sets of formulas.
    [Show full text]
  • Duality for Semiantichains and Unichain Coverings in Products of Special Posets
    Duality for semiantichains and unichain coverings in products of special posets Qi Liu and Douglas B. West∗ Abstract Saks and West conjectured that for every product of partial orders, the maximum size of a semiantichain equals the minimum number of unichains needed to cover the product. We prove the case where both factors have width 2. We also use the char- acterization of product graphs that are perfect to prove other special cases, including the case where both factors have height 2. Finally, we make some observations about the case where both factors have dimension 2. 1 Introduction In this paper, we study min-max relations involving natural subsets of partially ordered sets (posets). In a poset, chains and antichains are sets of pairwise comparable or pairwise incomparable elements, respectively. The height and width of a poset are the maximum sizes of chains and antichains, respectively. Since every chain contains at most one element from an antichain, the number of chains needed to cover the elements of a poset is at least its width. Dilworth [4] proved that this many chains suffice in any finite poset. A chain partition of a poset using the fewest chains is a Dilworth decomposition. Greene and Kleitman [7] generalized Dilworth's Theorem. A k-family in a poset is a family of elements containing no chain of size k + 1. Equivalently, a k-family is a union of k antichains. Every k-family has size at most minfk; jCijg, where C1; : : : ; Ct is a chain Pi decomposition, because chain Ci contributes at most minfk; jCijg elements to a k-family.
    [Show full text]
  • UNIVERSITДT BONN Physikalisches Institut
    DE05F5029 UNIVERSITÄT BONN Physikalisches Institut Families and Degenerations of Conformal Field Theories von Daniel Roggenkamp »DEO21241946* In this work, moduli spaces of conformal field theories are investigated. In the first part, moduli spaces corresponding to current-current defor- mation of conformal field theories are constructed explicitly. For WZW models, they are described in detail, and sigma model realizations of the deformed WZW models are presented. The second part is devoted to the study of boundaries of moduli spaces of conformal field theories. For this purpose a notion of convergence of families of conformal field theories is introduced, which admits certain degenerated conformal field theories to occur as limits. To. such a degeneration of conformal field the- ories, a degeneration of metric spaces together with additional geometric structures can be associated, which give rise to a geometric interpreta- tion. Boundaries of moduli spaces of toroidal conformal field theories, orbifolds thereof and WZW models are analyzed. Furthermore, also the limit of the discrete family of Virasoro minimal models is investigated. Post address: BONN-IR-2004-14 Nussallee 12 Bonn University- 53115 Bonn September 2004 Germany ISSN-0172-8741 Contents 1 Introduction 4 2 Moduli spaces of WZW models 8 2.1 Current-current deformations of CFTs 10 2.2 Some deformation theory 13 2.3 Deformations of WZW models 17 2.4 Sigma model description of deformed WZW models 20 2.4.1 Classical Action 20 2.4.2 Thedilaton 25 2.5 Explicit example: su(2)fc 31 2.6 Discussion 35 3 Limits and degenerations of CFTs 38 3.1 From geometry to CFT, and back to geometry 39 3.1.1 From Riemannian geometry to spectral triples 40 3.1.2 Spectral triples from CFTs 43 3.1.3 Commutative (sub)-geometries 47 3.2 Limits of conformal field theories: Definitions 52 3.2.1 Sequences of CFTs and their limits 52 3.2.2 Geometric interpretations 60 3.3 Limits of conformal field theories: Simple examples 64 3.3.1 Torus models 64 3.3.2 Torus orbifolds .
    [Show full text]
  • Duality Theory in Logic
    Duality Theory in Logic Sumit Sourabh ILLC, Universiteit van Amsterdam Cool Logic 14th December Duality in General “Duality underlines the world” “Most human things go in pairs” (Alcmaeon, 450 BC) • Existence of an entity in seemingly di↵erent forms, which are strongly related. Dualism forms a part of the philosophy of eastern religions. • In Physics : Wave-particle duality, electro-magnetic duality, • Quantum Physics,... Duality in Mathematics Back and forth mappings between dual classes of • mathematical objects. Lattices are self-dual objects • Projective Geometry • Vector Spaces • In logic, dualities have been used for relating syntactic and • semantic approaches. Algebras and Spaces Logic fits very well in between. Algebras Equational classes having a domain and operations . • Eg. Groups, Lattices, Boolean Algebras, Heyting Algebras ..., • Homomorphism, Subalgebras, Direct Products, Variety, ... • Power Set Lattice (BDL) BA = (BDL + )s.t.a a =1 • ¬ _ ¬ (Topological) Spaces Topology is the study of spaces. • A toplogy on a set X is a collection of subsets (open sets) of X , closed under arbitrary union and finite intersection. R 1 0 1 − Open Sets correspond to neighbourhoods of points in space. Metric topology, Product topology, Discrete topology ... • Continuous maps, Homeomorphisms, Connectedness, • Compactness, Hausdor↵ness,... “I dont consider this algebra, but this doesn’t mean that algebraists cant do it.” (Birkho↵) A brief history of Propositional Logic Boole’s The Laws of Thought • (1854) introduced an algebraic system for propositional
    [Show full text]
  • Descent and Duality
    ANNALS OF PURE AND APPLIED LOGIC ELSEVIER Annals of Pure and Applied Logic 71 (1995) 131-188 Descent and duality Marek W. Zawadowski Instytut Matematyki, Uniwersytet Warszawski, ul. Banachu 2, 00-913 Warmwa, Poland “The most interesting phenomena in model theory are the conclusions concerning the syntactical structure of a first order theory drawn from the examination of the models of the theory.” M. Makkai [15] Received 26 June 1992; revised 20 December 1993; communicated by D. van Dalen Abstract Using the Makkai’s duality for first-order logic, we characterise effective descent morphisms in 2-categories of prctoposes and Barr-exact categories. In both cases they coincide with conser- vative morphisms. We show that in those 2-categories the 2coregular factorisations are exactly quotient-conservative factorisations. We also prove a generalisation of the Makkai duality for pseudoelementary categories. 1. Introduction Consider the following well-known theorem about boolean algebras. Theorem 1.1. Let h : A - B be a homomorphism of boolean algebras. Then the pushout h A-B (1) IPI B-B+AB PO is a pullback (i.e. 0168-0072/95/$09.50 @ 1995 Elsevier Science B.V. All tights reserved SSDIO168-0072(94)00018-X 132 MU! Zawadowski/Annalsof Pure and Applied Logic 71 (1995) 131-188 h A-+-+ -B+AB PI is an equaliser) iff h is a monomolphism. Proof. One of the possible proofs goes as follows. We take the dual of (1) in the category of Stone spaces B* - po* (B+AW* Thus (2) is a pullback. Pullbacks and quotients of kernel pairs in the category of Stone spaces are computed in Sets.
    [Show full text]