Duality Theory in Logic
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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. -
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. -
Duality, Part 1: Dual Bases and Dual Maps Notation
Duality, part 1: Dual Bases and Dual Maps Notation F denotes either R or C. V and W denote vector spaces over F. Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Examples: Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). Examples: Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. -
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. -
Recent Developments in the Theory of Duality in Locally Convex Vector Spaces
[ VOLUME 6 I ISSUE 2 I APRIL– JUNE 2019] E ISSN 2348 –1269, PRINT ISSN 2349-5138 RECENT DEVELOPMENTS IN THE THEORY OF DUALITY IN LOCALLY CONVEX VECTOR SPACES CHETNA KUMARI1 & RABISH KUMAR2* 1Research Scholar, University Department of Mathematics, B. R. A. Bihar University, Muzaffarpur 2*Research Scholar, University Department of Mathematics T. M. B. University, Bhagalpur Received: February 19, 2019 Accepted: April 01, 2019 ABSTRACT: : The present paper concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Key Words: LOCALLY CONVEX SPACES We shall be exclusively concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A convex set A in a vector space E is symmetric if —A=A; then 0ЄA if A is not empty. A convex set A is absorbing if for every X≠0 in E), there exists a number α≠0 such that λxЄA for |λ| ≤ α ; this implies that A generates E. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Conversely, if any filter base is given on a vector space E, and consists of convex, symmetric, and absorbing sets, then it defines one and only one topology on E for which x+y and λx are continuous functions of both their arguments. -
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. -
Residues, Duality, and the Fundamental Class of a Scheme-Map
RESIDUES, DUALITY, AND THE FUNDAMENTAL CLASS OF A SCHEME-MAP JOSEPH LIPMAN Abstract. The duality theory of coherent sheaves on algebraic vari- eties goes back to Roch's half of the Riemann-Roch theorem for Riemann surfaces (1870s). In the 1950s, it grew into Serre duality on normal projective varieties; and shortly thereafter, into Grothendieck duality for arbitrary varieties and more generally, maps of noetherian schemes. This theory has found many applications in geometry and commutative algebra. We will sketch the theory in the reasonably accessible context of a variety V over a perfect field k, emphasizing the role of differential forms, as expressed locally via residues and globally via the fundamental class of V=k. (These notions will be explained.) As time permits, we will indicate some connections with Hochschild homology, and generalizations to maps of noetherian (formal) schemes. Even 50 years after the inception of Grothendieck's theory, some of these generalizations remain to be worked out. Contents Introduction1 1. Riemann-Roch and duality on curves2 2. Regular differentials on algebraic varieties4 3. Higher-dimensional residues6 4. Residues, integrals and duality7 5. Closing remarks8 References9 Introduction This talk, aimed at graduate students, will be about the duality theory of coherent sheaves on algebraic varieties, and a bit about its massive|and still continuing|development into Grothendieck duality theory, with a few indications of applications. The prerequisite is some understanding of what is meant by cohomology, both global and local, of such sheaves. Date: May 15, 2011. 2000 Mathematics Subject Classification. Primary 14F99. Key words and phrases. Hochschild homology, Grothendieck duality, fundamental class. -
SERRE DUALITY and APPLICATIONS Contents 1
SERRE DUALITY AND APPLICATIONS JUN HOU FUNG Abstract. We carefully develop the theory of Serre duality and dualizing sheaves. We differ from the approach in [12] in that the use of spectral se- quences and the Yoneda pairing are emphasized to put the proofs in a more systematic framework. As applications of the theory, we discuss the Riemann- Roch theorem for curves and Bott's theorem in representation theory (follow- ing [8]) using the algebraic-geometric machinery presented. Contents 1. Prerequisites 1 1.1. A crash course in sheaves and schemes 2 2. Serre duality theory 5 2.1. The cohomology of projective space 6 2.2. Twisted sheaves 9 2.3. The Yoneda pairing 10 2.4. Proof of theorem 2.1 12 2.5. The Grothendieck spectral sequence 13 2.6. Towards Grothendieck duality: dualizing sheaves 16 3. The Riemann-Roch theorem for curves 22 4. Bott's theorem 24 4.1. Statement and proof 24 4.2. Some facts from algebraic geometry 29 4.3. Proof of theorem 4.5 33 Acknowledgments 34 References 35 1. Prerequisites Studying algebraic geometry from the modern perspective now requires a some- what substantial background in commutative and homological algebra, and it would be impractical to go through the many definitions and theorems in a short paper. In any case, I can do no better than the usual treatments of the various subjects, for which references are provided in the bibliography. To a first approximation, this paper can be read in conjunction with chapter III of Hartshorne [12]. Also in the bibliography are a couple of texts on algebraic groups and Lie theory which may be beneficial to understanding the latter parts of this paper. -
Homotopy Theory and Generalized Duality for Spectral Sheaves
IMRN International Mathematics Research Notices 1996, No. 20 Homotopy Theory and Generalized Duality for Spectral Sheaves Jonathan Block and Andrey Lazarev We would like to express our sadness at the passing of Robert Thomason whose ideas have had a lasting impact on our work. The mathematical community is sorely impoverished by the loss. We dedicate this paper to his memory. 1 Introduction In this paper we announce a Verdier-type duality theorem for sheaves of spectra on a topological space X. Along the way we are led to develop the homotopy theory and stable homotopy theory of spectral sheaves. Such theories have been worked out in the past, most notably by [Br], [BG], [T], and [J]. But for our purposes these theories are inappropri- ate. They also have not been developed as fully as they are here. The previous authors did not have at their disposal the especially good categories of spectra constructed in [EKM], which allow one to do all of homological algebra in the category of spectra. Because we want to work in one of these categories of spectra, we are led to consider sheaves of spaces (as opposed to simplicial sets), and this gives rise to some additional technical difficulties. As an application we compute an example from geometric topology of stratified spaces. In future work we intend to apply our theory to equivariant K-theory. 2 Spectral sheaves There are numerous categories of spectra in existence and our theory can be developed in a number of them. Received 4 June 1996. Revision received 9 August 1996. -
Duality for Mixed-Integer Linear Programs
Duality for Mixed-Integer Linear Programs M. Guzelsoy∗ T. K. Ralphs† Original May, 2006 Revised April, 2007 Abstract The theory of duality for linear programs is well-developed and has been successful in advancing both the theory and practice of linear programming. In principle, much of this broad framework can be ex- tended to mixed-integer linear programs, but this has proven difficult, in part because duality theory does not integrate well with current computational practice. This paper surveys what is known about duality for integer programs and offers some minor extensions, with an eye towards developing a more practical framework. Keywords: Duality, Mixed-Integer Linear Programming, Value Function, Branch and Cut. 1 Introduction Duality has long been a central tool in the development of both optimization theory and methodology. The study of duality has led to efficient procedures for computing bounds, is central to our ability to perform post facto solution analysis, is the basis for procedures such as column generation and reduced cost fixing, and has yielded optimality conditions that can be used as the basis for “warm starting” techniques. Such procedures are useful both in cases where the input data are subject to fluctuation after the solution procedure has been initiated and in applications for which the solution of a series of closely-related instances is required. This is the case for a variety of integer optimization algorithms, including decomposition algorithms, parametric and stochastic programming algorithms, multi-objective optimization algorithms, and algorithms for analyzing infeasible mathematical models. The theory of duality for linear programs (LPs) is well-developed and has been extremely successful in contributing to both theory and practice. -
Duality in MIP
Duality in MIP Generating Dual Price Functions Using Branch-and-Cut Elena V. Pachkova Abstract This presentation treats duality in Mixed Integer Programming (MIP in short). A dual of a MIP problem includes a dual price function F , that plays the same role as the dual variables in Linear Programming (LP in the following). The price function is generated while solving the primal problem. However, di®erent to the LP dual variables, the characteristics of the dual price function depend on the algorithmic approach used to solve the MIP problem. Thus, the cutting plane approach provides non- decreasing and superadditive price functions while branch-and-bound algorithm generates piecewise linear, nondecreasing and convex price functions. Here a hybrid algorithm based on branch-and-cut is investigated, and a price function for that algorithm is established. This price function presents a generalization of the dual price functions obtained by either the cutting plane or the branch-and-bound method. 1 Introduction Duality in mathematical programming is used in a variety of applications. Apart from con- ceptual interest it provides interesting economic interpretations of the problem. Moreover, using dual information usually improves the performance of an algorithm. Thus, there exist many results on duality in linear programming (LP) (e.g. see Gass (1985)). Results on duality in integer programming (IP) also exist (Wolsey (1981)). While algorithms for LP produce unique dual programs (apart from degenerating programs), that are relatively easy to obtain, IP algorithms generate a dual function whose characteristics depend on the method used to solve the primal IP problem. Wolsey (1981) characterized this function for 73 the branch-and-bound and the cutting plane methods. -
SCUOLA NORMALE SUPERIORE DI PISA Classe Di Scienze
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze ALDO ANDREOTTI ARNOLD KAS Duality on complex spaces Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 27, no 2 (1973), p. 187-263 <http://www.numdam.org/item?id=ASNSP_1973_3_27_2_187_0> © Scuola Normale Superiore, Pisa, 1973, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ DUALITY ON COMPLEX SPACES by ALDO ANDREOTTI and ARNOLD KAS This paper has grown out of a seminar held at Stanford by us on the subject of Serre duality [20]. It should still be regarded as a seminar rather than as an original piece of research. The method of presentation is as elementary as possible with the intention of making the results available and understandable to the non-specialist. Perhaps the main feature is the introduction of Oech homology; the duality theorem is essentially divided into two steps, first duality between cohomology and homology and then an algebraic part expressing the homology groups in terms of well-known functors. We have wished to keep these steps separated as it is only in the first part that the theory of topological vector spaces is of importance.