MV-algebras and fuzzy topologies: Stone duality extended
|
|
- Angela Floyd
- 5 years ago
- Views:
Transcription
1 MV-algebras and fuzzy topologies: Stone duality extended Dipartimento di Matematica Università di Salerno, Italy Algebra and Coalgebra meet Proof Theory Universität Bern April 27 29, 2011
2 Outline 1 MV-algebras and their reducts 2 3 4
3 Outline MV-algebras and their reducts MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras 1 MV-algebras and their reducts 2 3 4
4 MV-algebras MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras Definition An MV-algebra A,,, 0 is an algebra of type (2,1,0) such that A,, 0 is a commutative monoid, (x ) = x, x 0 = 0, (x y) y = (y x) x. The MV-algebra [0, 1] [0, 1],,, 0, with x y := min{x + y, 1} and x := 1 x, is an MV-algebra, called standard. It generates the variety of MV-algebras both as a variety and as a quasi-variety.
5 Further operations and properties MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras Operations x y if and only if x y = 1, 1 = 0, x y = (x y ), defines a structure of bounded lattice. Properties, and distribute over any existing join., and distribute over any existing meet. De Morgan laws hold both for weak and strong conjunction and disjunction: x y = (x y ) and x y = (x y ), x y = (x y ) and x y = (x y ).
6 MV and Boolean algebras MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras Boole MV Boolean algebras form a subvariety of the variety of MV-algebras. They are the MV-algebras satisfying the equation x x = x. The Boolean center Let A be an MV-algebra. a A is called idempotent or Boolean if a a = a. a a = a iff a a = a. a is Boolean iff a is. B(A) = {a A a a = a} is a Boolean algebra, called the Boolean center of A. It is, in fact, the largest Boolean subalgebra of A.
7 Semirings and quantales MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras Definition A semiring is a structure S, +,, 0 such that S, +, 0 is a commutative monoid, S, is a semigroup, distributes over + from either side. Definition A quantale Q,,, is a sup-lattice equipped with a monoid operation which distributes over arbitrary joins.
8 Reducts of MV-algebras MV-algebras MV and Boolean algebras Semiring and quantale reducts of MV-algebras [Di Nola Gerla B., 2005] For any MV-algebra A, A,,, 0, 1 and A,,, 1, 0 are (commutative, unital, additively idempotent) semirings, isomorphic under the negation. So, if A is complete, A,,, 0, 1 and A,,, 1, 0 are isomorphic (commutative, unital) quantales. Moreover, also A,,, 0 and A,,, 1 are isomorphic semirings and, if A is complete, A,,, 0 and A,,, 1 are isomorphic quantales.
9 Outline MV-algebras and their reducts Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras 1 MV-algebras and their reducts 2 3 4
10 Ideals and congruences in MV Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Definition A subset I of an MV-algebra A is called an ideal if 0 I ; I is downward closed; a b I for all a, b I. Proposition Let A be an MV-algebra. For any MV-algebra congruence on A, [0] is an ideal of A. Conversely, for any ideal I, the relation I defined by a I b iff d(a, b) := (a b ) (b a ) I is the unique congruence on A whose zero-class is I.
11 Maximal ideals Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Max A The set of all maximal ideals of A is denoted by Max A. The radical of A is defined as the intersection of all maximal ideals: Rad A := Max A. Proposition If M is a proper ideal of A then the following are equivalent: M is maximal; for any a A, if a / M then there exists n ω such that (a ) n M.
12 Semisimple algebras Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Definition (from Universal Algebra) An algebra A is called semisimple if it is subdirect product of simple algebras. Proposition An MV-algebra A is semisimple if and only if Rad A = {0}. MV ss The class of semisimple MV-algebras form a full subcategory of MV that we shall denote by MV ss. It is worth noticing that, although MV ss is NOT a variety (it is closed under S and P, but not under H), it contains [0, 1], Boole, and free, projective, σ-complete and complete MV-algebras.
13 Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Semisimple MV-algebras are algebras of fuzzy sets Theorem [Belluce, 1986] A is isomorphic to a subalgebra of [0, 1] Max A, for any A MV ss. Sketch of the proof. For any M Max A, A/M is simple. [Chang, 1959]: Any simple MV-algebra is an archimedean chain, hence it is isomorphic to a (unique) subalgebra of [0, 1]. So there exists a unique embedding ι M : A/M [0, 1]. Let ϕ M : A A/M be the natural projection. a A, let â : M Max A ι M (ϕ M (a)) [0, 1]. The map ι : a A â [0, 1] Max A is an MV-algebra embedding.
14 Hyperarchimedean algebras Maximal ideals Semisimple algebras Belluce theorem Hyperarchimedean algebras Definition Let A be an MV-algebra. An element a A is archimedean if it satisfies the following equivalent conditions: 1 there exists a positive integer n such that na B(A); 2 there exists a positive integer n such that a na = 1; 3 there exists a positive integer n such that na = (n + 1)a. Definition An MV-algebra A is called hyperarchimedean if all of its elements are archimedean.
15 Outline MV-algebras and their reducts The category MV Top The shadow topology 1 MV-algebras and their reducts 2 3 4
16 Open sets MV-algebras and their reducts The category MV Top The shadow topology X, Ω topological space {0, 1} X,,,, 0, 1 is a complete Boolean algebra. Ω,, 0 is a sup-sublattice of {0, 1} X,, 0, Ω,, 1 is a meet-subsemilattice of {0, 1} X,, 1. X, Ω MV-topological space [0, 1] X,,,,,, 0, 1 is a complete MV-algebra. Ω,,, 0 is a subquantale of [0, 1] X,,, 0, Ω,,, 1 is a subsemiring of [0, 1] X,,, 1.
17 Continuous maps The category MV Top The shadow topology Preimage of a function Let X, Y be sets and f : X Y a map. If we identify the subsets of X and Y with their membership functions, the preimage of f is f : χ {0, 1} Y χ f {0, 1} X. Analogously, the fuzzy preimage of f is defined by f : χ [0, 1] Y χ f [0, 1] X. MV-continuity So, if X, Ω X and Y, Ω Y are MV-spaces, f : X Y is said to be MV-continuous if f [Ω Y ] Ω X.
18 Examples and bases The category MV Top The shadow topology X, {0, 1} and X, [0, 1] X are MV-topological spaces. Any topology is an MV-topology. Let d : X [0, + [ be a metric on X and α a fuzzy point of X with support { x. For any r R +, the open ball B r (α) is α(x) if d(x, y) < r B r (α)(y) := 0 if d(x, y) r. The family of fuzzy subsets of X that are joins of open balls is an MV-topology on X that is said to be induced by d. Definition T = X, Ω MV Top. B Ω is called a base for T if, for all o Ω, o = i I b i, with {b i } i I B.
19 The shadow topology The category MV Top The shadow topology Definition For any MV-space T = X, Ω, let B(Ω) := Ω {0, 1} X. Sh T = X, B(Ω) is a topology in the classical sense, called the shadow of T. Sh is a functor Top is a full subcategory of MV Top. The mapping Sh : MV Top Top is a functor. It is, in fact, the left-inverse of the inclusion Top MV Top. The shadow of the MV-topology induced by a metric d is the topology induced by d.
20 Outline MV-algebras and their reducts Stone MV-spaces Stone duality extended Some compositions 1 MV-algebras and their reducts 2 3 4
21 Compactness Stone MV-spaces Stone duality extended Some compositions A more complex situation Due to the presence of two intersection and two union operations, compactness and each separation axiom can have at least two different MV-versions. Compact spaces An MV-space X, Ω is said to be compact if any open covering of X contains an additive covering, i.e., for any Ω Ω such that Ω = 1, there exists a finite subset {o 1,..., o n } of Ω such that o 1 o n = 1; strongly compact if any open covering of X contains a finite covering.
22 Separation MV-algebras and their reducts Stone MV-spaces Stone duality extended Some compositions T 2 axioms An MV-space T = X, Ω is called an Hausdorff (or separated, or T 2 ) space if, for any x y X, there exist o x, o y Ω such that: (i) o x (x) = o y (y) = 1, (ii) o x (y) = o y (x) = 0, (iii) o x o y = 0. T is said to be strongly separated if, for any x y X, there exist o x, o y Ω satisfying (i) and (iv) o x o y = 0. T 2 definition do not need fuzzy points.
23 Stone MV-spaces Stone MV-spaces Stone duality extended Some compositions Remark Strong separation implies separation and they both collapse to classical T 2 in the case of crisp topologies. The same holds for compactness. Clopens and zero-dimensionality Let T = X, Ω be an MV-space and Ξ = Ω be the family of closed fuzzy subsets. We denote by Clop T the family Ω Ξ of clopen fuzzy subsets of X. Clop T MV ss, for any MV-space T. T is called zero-dimensional if Clop T is a base for it. Definition A Stone MV-space is an MV-space which is compact, separated and zero-dimensional.
24 The MV-space Max A, Ω A Stone MV-spaces Stone duality extended Some compositions Remark The category MV Stone of Stone MV-spaces, with MV-continuous maps as morphisms, is a full subcategory of MV Top. The Maximal MV-spectrum Let A be a semisimple MV-algebra. By Belluce representation theorem, there exists a canonical embedding ι : A [0, 1] Max A. Then ι[a] generates, as a base, an MV-topology on Max A. The family of open sets of such a space is denoted by Ω A. So, for any semisimple MV-algebra A, Max A, Ω A denotes the MV-topological space on Max A having (an isomorphic copy of) A as a base.
25 Stone MV-spaces Stone duality extended Some compositions A (proper) extension of Stone duality Theorem 1 The mappings Φ : T MV Top Clop T MV ss Ψ : A MV ss Max A, Ω A MV Top define two contravariant functors. 2 They yield a duality between MV ss and MV Stone, that is for every semisimple MV-algebra A, ΨA is a Stone MV-space and A is isomorphic to the clopen algebra of such a space; conversely, every Stone MV-space T = X, Ω is homeomorphic to ΨΦT. 3 The restriction of such a duality to Boolean algebras and Stone spaces coincide with the classical Stone duality. 4 Φ Sh = B Φ and Ψ B = Sh Ψ.
26 MV-algebras and their reducts Graphically Stone MV-spaces Stone duality extended Some compositions MV ss Ψ MV Stone op Φ B Sh Boole Φ Ψ Stone op Horizontal arrows: equivalences Vertical arrows: inclusions of full subcategories and their left-inverses Corollary Strongly separated Stone MV-spaces are dual to hyperarchimedean MV-algebras.
27 Mundici equivalence Stone MV-spaces Stone duality extended Some compositions Unital Abelian l-groups Let ulg Ab be the category whose objects are Abelian lattice-ordered groups with a distinguished strong order unit and whose morphisms are unit-preserving l-group homomorphisms. Theorem [Mundici, 1986] The categories ulg Ab and MV are equivalent. Γ : ulg Ab MV is defined as follows: for any Abelian ul-group G, +,,,, 0, u, Γ(G) = [0, u],,, 0, where x y := (x + y) u, x = u x. We shall denote by Γ 1 the inverse of Γ.
28 ΦΓ and Γ 1 Ψ Stone MV-spaces Stone duality extended Some compositions Restrictions of Γ and Γ 1 MV ss is equivalent to the subcategory ulgr Ab of ulgab whose objects are, up to isomorphisms, ul-subgroups of a group of bounded real-valued functions from a set X, with 1 as unit. In such a restriction Boolean algebras correspond to subgroups of bounded integer-valued functions (ulg Ab Z ). Corollary Then ΦΓ and Γ 1 Ψ form a duality between ulgr Ab and MV Stone. Such a duality obviously restricts to ulgz Ab and Stone.
29 n-valued MV-algebras Stone MV-spaces Stone duality extended Some compositions MV n and BR n Let n > 1 be a natural number. In [Di Nola Lettieri, 2000], the authors defined the category BR n : objects are pairs (B, R), where B is a Boolean algebra and R is an n-ary relation on B satisfying certain conditions; a morphism from (B, R) to (B, R ) is a Boolean algebra homomorphism f : B B such that (a 1,..., a n ) R implies (f (a 1 ),..., f (a n )) R. Now, let MV n denote the subvariety V(S n ) of MV generated by the (n + 1)-element chain S n = {0, 1 n,..., n 1 n, 1}.
30 Future work Stone MV-spaces Stone duality extended Some compositions Theorem [Di Nola Lettieri, 2000] The categories MV n and BR n are equivalent. MV n MV ss, for all n > 1. MV Stone and Stone From an MV-topological viewpoint, MV n is dual to the category of Stone MV-spaces of fuzzy sets with S n -valued membership functions. Next step will be to characterize a suitable category of Stone spaces with additional conditions which is dual to BR n and, therefore, to MV n.
31 Further possible developments Stone MV-spaces Stone duality extended Some compositions A point-free approach MV-frames (reducts of complete BL-algebras?). Spatial and sober MV-frames. MV-topoi. Applications Mathematical Morphology in digital image analysis. Geosystems. Other ideas or suggestions are welcome...
32 Stone MV-spaces Stone duality extended Some compositions Thank you!
33 Stone MV-spaces Stone duality extended Some compositions References [Belluce, 1986], Semisimple algebras of infinite valued logic and bold fuzzy set theory, Can. J. Math., 38/6, , [Chang, 1959], A new proof of the completeness of the Lukasiewicz axioms, Trans. Amer. Math. Soc., 93, 74 90, [Di Nola Gerla B., 2005], Algebras of Lukasiewicz s logic and their semiring reducts, Contemp. Math., 377, , Amer. Math. Soc., [Di Nola Lettieri, 2000], One chain generated varieties of MV-algebras, Journal of Algebra, 225, , [Mundici, 1986] Interpretation of AF C -algebras in Lukasiewicz sentential calculus, J. Functional Analysis, 65, 15 63, [Russo], An extension of Stone duality to fuzzy topologies and MV-algebras, submitted, arxiv: v2 [math.lo].
The logic of perfect MV-algebras
The logic of perfect MV-algebras L. P. Belluce Department of Mathematics, British Columbia University, Vancouver, B.C. Canada. belluce@math.ubc.ca A. Di Nola DMI University of Salerno, Salerno, Italy adinola@unisa.it
More informationThe prime spectrum of MV-algebras based on a joint work with A. Di Nola and P. Belluce
The prime spectrum of MV-algebras based on a joint work with A. Di Nola and P. Belluce Luca Spada Department of Mathematics and Computer Science University of Salerno www.logica.dmi.unisa.it/lucaspada
More informationMorita-equivalences for MV-algebras
Morita-equivalences for MV-algebras Olivia Caramello* University of Insubria Geometry and non-classical logics 5-8 September 2017 *Joint work with Anna Carla Russo O. Caramello Morita-equivalences for
More informationBoolean Algebras, Boolean Rings and Stone s Representation Theorem
Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to
More informationAn algebraic approach to Gelfand Duality
An algebraic approach to Gelfand Duality Guram Bezhanishvili New Mexico State University Joint work with Patrick J Morandi and Bruce Olberding Stone = zero-dimensional compact Hausdorff spaces and continuous
More informationGeometric aspects of MV-algebras. Luca Spada Università di Salerno
Geometric aspects of MV-algebras Luca Spada Università di Salerno TACL 2017 TACL 2003 Tbilisi, Georgia. Contents Crash tutorial on MV-algebras. Dualities for semisimple MV-algebras. Non semisimple MV-algebras.
More informationThe Blok-Ferreirim theorem for normal GBL-algebras and its application
The Blok-Ferreirim theorem for normal GBL-algebras and its application Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics
More informationDistributive Lattices with Quantifier: Topological Representation
Chapter 8 Distributive Lattices with Quantifier: Topological Representation Nick Bezhanishvili Department of Foundations of Mathematics, Tbilisi State University E-mail: nickbezhanishvilli@netscape.net
More informationThe Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Algebras of Lukasiewicz s Logic and their Semiring Reducts A. Di Nola B. Gerla Vienna, Preprint
More informationEmbedding theorems for normal divisible residuated lattices
Embedding theorems for normal divisible residuated lattices Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics and Computer
More informationMV -ALGEBRAS ARE CATEGORICALLY EQUIVALENT TO A CLASS OF DRl 1(i) -SEMIGROUPS. (Received August 27, 1997)
123 (1998) MATHEMATICA BOHEMICA No. 4, 437 441 MV -ALGEBRAS ARE CATEGORICALLY EQUIVALENT TO A CLASS OF DRl 1(i) -SEMIGROUPS Jiří Rachůnek, Olomouc (Received August 27, 1997) Abstract. In the paper it is
More informationCongruence Boolean Lifting Property
Congruence Boolean Lifting Property George GEORGESCU and Claudia MUREŞAN University of Bucharest Faculty of Mathematics and Computer Science Academiei 14, RO 010014, Bucharest, Romania Emails: georgescu.capreni@yahoo.com;
More information2. ETALE GROUPOIDS MARK V. LAWSON
2. ETALE GROUPOIDS MARK V. LAWSON Abstract. In this article, we define étale groupoids and describe some of their properties. 1. Generalities 1.1. Categories. A category is usually regarded as a category
More informationCategory Theory (UMV/TK/07)
P. J. Šafárik University, Faculty of Science, Košice Project 2005/NP1-051 11230100466 Basic information Extent: 2 hrs lecture/1 hrs seminar per week. Assessment: Written tests during the semester, written
More informationMonadic GMV -algebras
Department of Algebra and Geometry Faculty of Sciences Palacký University of Olomouc Czech Republic TANCL 07, Oxford 2007 monadic structures = algebras with quantifiers = algebraic models for one-variable
More informationAlgebra and probability in Lukasiewicz logic
Algebra and probability in Lukasiewicz logic Ioana Leuştean Faculty of Mathematics and Computer Science University of Bucharest Probability, Uncertainty and Rationality Certosa di Pontignano (Siena), 1-3
More informationSummary of the PhD Thesis MV-algebras with products: connecting the Pierce-Birkhoff conjecture with Lukasiewicz logic Serafina Lapenta
Summary of the PhD Thesis MV-algebras with products: connecting the Pierce-Birkhoff conjecture with Lukasiewicz logic Serafina Lapenta The framework. In 1956, G. Birkhoff G. and R.S. Pierce [1] conjectured
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato September 10, 2015 ABSTRACT. This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a
More informationVarieties of Heyting algebras and superintuitionistic logics
Varieties of Heyting algebras and superintuitionistic logics Nick Bezhanishvili Institute for Logic, Language and Computation University of Amsterdam http://www.phil.uu.nl/~bezhanishvili email: N.Bezhanishvili@uva.nl
More informationOn the filter theory of residuated lattices
On the filter theory of residuated lattices Jiří Rachůnek and Dana Šalounová Palacký University in Olomouc VŠB Technical University of Ostrava Czech Republic Orange, August 5, 2013 J. Rachůnek, D. Šalounová
More informationNotes about Filters. Samuel Mimram. December 6, 2012
Notes about Filters Samuel Mimram December 6, 2012 1 Filters and ultrafilters Definition 1. A filter F on a poset (L, ) is a subset of L which is upwardclosed and downward-directed (= is a filter-base):
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato June 23, 2015 This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection
More informationINTRODUCING MV-ALGEBRAS. Daniele Mundici
INTRODUCING MV-ALGEBRAS Daniele Mundici Contents Chapter 1. Chang subdirect representation 5 1. MV-algebras 5 2. Homomorphisms and ideals 8 3. Chang subdirect representation theorem 11 4. MV-equations
More informationAndrzej Walendziak, Magdalena Wojciechowska-Rysiawa BIPARTITE PSEUDO-BL ALGEBRAS
DEMONSTRATIO MATHEMATICA Vol. XLIII No 3 2010 Andrzej Walendziak, Magdalena Wojciechowska-Rysiawa BIPARTITE PSEUDO-BL ALGEBRAS Abstract. The class of bipartite pseudo-bl algebras (denoted by BP) and the
More informationSymbol Index Group GermAnal Ring AbMonoid
Symbol Index 409 Symbol Index Symbols of standard and uncontroversial usage are generally not included here. As in the word index, boldface page-numbers indicate pages where definitions are given. If a
More informationA VIEW OF CANONICAL EXTENSION
A VIEW OF CANONICAL EXTENSION MAI GEHRKE AND JACOB VOSMAER Abstract. This is a short survey illustrating some of the essential aspects of the theory of canonical extensions. In addition some topological
More informationComputing Spectra via Dualities in the MTL hierarchy
Computing Spectra via Dualities in the MTL hierarchy Diego Valota Department of Computer Science University of Milan valota@di.unimi.it 11th ANNUAL CECAT WORKSHOP IN POINTFREE MATHEMATICS Overview Spectra
More informationProbability Measures in Gödel Logic
Probability Measures in Gödel Logic Diego Valota Department of Computer Science University of Milan valota@di.unimi.it Joint work with Stefano Aguzzoli (UNIMI), Brunella Gerla and Matteo Bianchi (UNINSUBRIA)
More informationSkew Boolean algebras
Skew Boolean algebras Ganna Kudryavtseva University of Ljubljana Faculty of Civil and Geodetic Engineering IMFM, Ljubljana IJS, Ljubljana New directions in inverse semigroups Ottawa, June 2016 Plan of
More informationFuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras
Fuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras Jiří Rachůnek 1 Dana Šalounová2 1 Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova
More informationAn adjoint construction for topological models of intuitionistic modal logic Extended abstract
An adjoint construction for topological models of intuitionistic modal logic Extended abstract M.J. Collinson, B.P. Hilken, D.E. Rydeheard April 2003 The purpose of this paper is to investigate topological
More informationLogic Synthesis and Verification
Logic Synthesis and Verification Boolean Algebra Jie-Hong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 2014 1 2 Boolean Algebra Reading F. M. Brown. Boolean Reasoning:
More informationContents. Introduction
Contents Introduction iii Chapter 1. Residuated lattices 1 1. Definitions and preliminaries 1 2. Boolean center of a residuated lattice 10 3. The lattice of deductive systems of a residuated lattice 14
More informationCATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS. Contents. 1. The ring K(R) and the group Pic(R)
CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS J. P. MAY Contents 1. The ring K(R) and the group Pic(R) 1 2. Symmetric monoidal categories, K(C), and Pic(C) 2 3. The unit endomorphism ring R(C ) 5 4.
More informationON THE CONGRUENCE LATTICE OF A FRAME
PACIFIC JOURNAL OF MATHEMATICS Vol. 130, No. 2,1987 ON THE CONGRUENCE LATTICE OF A FRAME B. BANASCHEWSKI, J. L. FRITH AND C. R. A. GILMOUR Recall that the Skula modification SkX of a topological space
More informationA fresh perspective on canonical extensions for bounded lattices
A fresh perspective on canonical extensions for bounded lattices Mathematical Institute, University of Oxford Department of Mathematics, Matej Bel University Second International Conference on Order, Algebra
More informationAN ALGEBRAIC APPROACH TO GENERALIZED MEASURES OF INFORMATION
AN ALGEBRAIC APPROACH TO GENERALIZED MEASURES OF INFORMATION Daniel Halpern-Leistner 6/20/08 Abstract. I propose an algebraic framework in which to study measures of information. One immediate consequence
More informationSome consequences of compactness in Lukasiewicz Predicate Logic
Some consequences of compactness in Lukasiewicz Predicate Logic Luca Spada Department of Mathematics and Computer Science University of Salerno www.logica.dmi.unisa.it/lucaspada 7 th Panhellenic Logic
More informationModel Theory and Modules, Granada
Model Theory and Modules, Granada Mike Prest November 2015 Contents 1 Model Theory 1 2 Model Theory of Modules 4 2.1 Duality......................................... 7 3 Ziegler Spectra and Definable Categories
More informationA duality-theoretic approach to MTL-algebras
A duality-theoretic approach to MTL-algebras Sara Ugolini (Joint work with W. Fussner) BLAST 2018 - Denver, August 6th 2018 A commutative, integral residuated lattice, or CIRL, is a structure A = (A,,,,,
More informationTopological algebra based on sorts and properties as free and cofree universes
Topological algebra based on sorts and properties as free and cofree universes Vaughan Pratt Stanford University BLAST 2010 CU Boulder June 2 MOTIVATION A well-structured category C should satisfy WS1-5.
More informationPeter Hochs. Strings JC, 11 June, C -algebras and K-theory. Peter Hochs. Introduction. C -algebras. Group. C -algebras.
and of and Strings JC, 11 June, 2013 and of 1 2 3 4 5 of and of and Idea of 1 Study locally compact Hausdorff topological spaces through their algebras of continuous functions. The product on this algebra
More informationA topological duality for posets
A topological duality for posets RAMON JANSANA Universitat de Barcelona join work with Luciano González TACL 2015 Ischia, June 26, 2015. R. Jansana 1 / 20 Introduction In 2014 M. A. Moshier and P. Jipsen
More informationOn injective constructions of S-semigroups. Jan Paseka Masaryk University
On injective constructions of S-semigroups Jan Paseka Masaryk University Joint work with Xia Zhang South China Normal University BLAST 2018 University of Denver, Denver, USA Jan Paseka (MU) 10. 8. 2018
More informationEilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )
II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups
More informationP.S. Gevorgyan and S.D. Iliadis. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 70, 2 (208), 0 9 June 208 research paper originalni nauqni rad GROUPS OF GENERALIZED ISOTOPIES AND GENERALIZED G-SPACES P.S. Gevorgyan and S.D. Iliadis Abstract. The
More informationACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes)
ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes) Steve Vickers CS Theory Group Birmingham 2. Theories and models Categorical approach to many-sorted
More informationUniversal Algebra for Logics
Universal Algebra for Logics Joanna GRYGIEL University of Czestochowa Poland j.grygiel@ajd.czest.pl 2005 These notes form Lecture Notes of a short course which I will give at 1st School on Universal Logic
More informationA Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries
A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries Johannes Marti and Riccardo Pinosio Draft from April 5, 2018 Abstract In this paper we present a duality between nonmonotonic
More informationDuality and Automata Theory
Duality and Automata Theory Mai Gehrke Université Paris VII and CNRS Joint work with Serge Grigorieff and Jean-Éric Pin Elements of automata theory A finite automaton a 1 2 b b a 3 a, b The states are
More informationMonadic MV-algebras I: a study of subvarieties
Algebra Univers. 71 (2014) 71 100 DOI 10.1007/s00012-014-0266-3 Published online January 22, 2014 Springer Basel 2014 Algebra Universalis Monadic MV-algebras I: a study of subvarieties Cecilia R. Cimadamore
More informationCompactifications of Discrete Spaces
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 22, 1079-1084 Compactifications of Discrete Spaces U. M. Swamy umswamy@yahoo.com Ch. Santhi Sundar Raj, B. Venkateswarlu and S. Ramesh Department of Mathematics
More informationLattices, closure operators, and Galois connections.
125 Chapter 5. Lattices, closure operators, and Galois connections. 5.1. Semilattices and lattices. Many of the partially ordered sets P we have seen have a further valuable property: that for any two
More informationCOUNTABLE CHAINS OF DISTRIBUTIVE LATTICES AS MAXIMAL SEMILATTICE QUOTIENTS OF POSITIVE CONES OF DIMENSION GROUPS
COUNTABLE CHAINS OF DISTRIBUTIVE LATTICES AS MAXIMAL SEMILATTICE QUOTIENTS OF POSITIVE CONES OF DIMENSION GROUPS PAVEL RŮŽIČKA Abstract. We construct a countable chain of Boolean semilattices, with all
More informationDuality, Residues, Fundamental class
Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class
More information1 Differentiable manifolds and smooth maps
1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set
More informationCHAPTER 4. βs as a semigroup
CHAPTER 4 βs as a semigroup In this chapter, we assume that (S, ) is an arbitrary semigroup, equipped with the discrete topology. As explained in Chapter 3, we will consider S as a (dense ) subset of its
More informationMathematica Bohemica
Mathematica Bohemica Roman Frič Extension of measures: a categorical approach Mathematica Bohemica, Vol. 130 (2005), No. 4, 397 407 Persistent URL: http://dml.cz/dmlcz/134212 Terms of use: Institute of
More informationLattice Theory Lecture 5. Completions
Lattice Theory Lecture 5 Completions John Harding New Mexico State University www.math.nmsu.edu/ JohnHarding.html jharding@nmsu.edu Toulouse, July 2017 Completions Definition A completion of a poset P
More informationSets and Motivation for Boolean algebra
SET THEORY Basic concepts Notations Subset Algebra of sets The power set Ordered pairs and Cartesian product Relations on sets Types of relations and their properties Relational matrix and the graph of
More informationPriestley Duality for Bilattices
A. Jung U. Rivieccio Priestley Duality for Bilattices In memoriam Leo Esakia Abstract. We develop a Priestley-style duality theory for different classes of algebras having a bilattice reduct. A similar
More informationRINGS IN POST ALGEBRAS. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXVI, 2(2007), pp. 263 272 263 RINGS IN POST ALGEBRAS S. RUDEANU Abstract. Serfati [7] defined a ring structure on every Post algebra. In this paper we determine all the
More informationTopological aspects of restriction categories
Calgary 2006, Topological aspects of restriction categories, June 1, 2006 p. 1/22 Topological aspects of restriction categories Robin Cockett robin@cpsc.ucalgary.ca University of Calgary Calgary 2006,
More informationON FUZZY IDEALS OF PSEUDO MV -ALGEBRAS
Discussiones Mathematicae General Algebra and Applications 28 (2008 ) 63 75 ON FUZZY IDEALS OF PSEUDO MV -ALGEBRAS Grzegorz Dymek Institute of Mathematics and Physics University of Podlasie 3 Maja 54,
More informationEQUIVALENCE RELATIONS AND OPERATORS ON ORDERED ALGEBRAIC STRUCTURES. UNIVERSITÀ DEGLI STUDI DELL'INSUBRIA Via Ravasi 2, Varese, Italy
UNIVERSITÀ DEGLI STUDI DELL'INSUBRIA Via Ravasi 2, 21100 Varese, Italy Dipartimento di Scienze Teoriche e Applicate Di.S.T.A. Dipartimento di Scienza e Alta Tecnologia Di.S.A.T. PH.D. DEGREE PROGRAM IN
More informationUniversal Properties
A categorical look at undergraduate algebra and topology Julia Goedecke Newnham College 24 February 2017, Archimedeans Julia Goedecke (Newnham) 24/02/2017 1 / 30 1 Maths is Abstraction : more abstraction
More informationObstinate filters in residuated lattices
Bull. Math. Soc. Sci. Math. Roumanie Tome 55(103) No. 4, 2012, 413 422 Obstinate filters in residuated lattices by Arsham Borumand Saeid and Manijeh Pourkhatoun Abstract In this paper we introduce the
More informationAdvances in the theory of fixed points in many-valued logics
Advances in the theory of fixed points in many-valued logics Department of Mathematics and Computer Science. Università degli Studi di Salerno www.logica.dmi.unisa.it/lucaspada 8 th International Tbilisi
More informationSPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM
SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM SERGIO A. CELANI AND MARÍA ESTEBAN Abstract. Distributive Hilbert Algebras with infimum, or DH -algebras, are algebras with implication
More informationCATEGORY THEORY. Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths.
CATEGORY THEORY PROFESSOR PETER JOHNSTONE Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths. Definition 1.1. A category C consists
More informationLectures on étale groupoids, inverse semigroups and quantales
Lectures on étale groupoids, inverse semigroups and quantales Pedro Resende 1 August 2006 1 Research supported in part by FCT through the Program POCI2010/FEDER and by SOCRATES. Preface Groupoids and
More informationPartially ordered monads and powerset Kleene algebras
Partially ordered monads and powerset Kleene algebras Patrik Eklund 1 and Werner Gähler 2 1 Umeå University, Department of Computing Science, SE-90187 Umeå, Sweden peklund@cs.umu.se 2 Scheibenbergstr.
More informationA locally finite characterization of AE(0) and related classes of compacta
A locally finite characterization of AE(0) and related classes of compacta David Milovich Texas A&M International University david.milovich@tamiu.edu http://www.tamiu.edu/ dmilovich/ March 13, 2014 Spring
More informationThe lattice of varieties generated by residuated lattices of size up to 5
The lattice of varieties generated by residuated lattices of size up to 5 Peter Jipsen Chapman University Dedicated to Hiroakira Ono on the occasion of his 7th birthday Introduction Residuated lattices
More informationOn k-groups and Tychonoff k R -spaces (Category theory for topologists, topology for group theorists, and group theory for categorical topologists)
On k-groups and Tychonoff k R -spaces 0 On k-groups and Tychonoff k R -spaces (Category theory for topologists, topology for group theorists, and group theory for categorical topologists) Gábor Lukács
More informationThe Morita-equivalence between MV-algebras and abelian l-groups with strong unit
The Morita-equivalence between MV-algebras and abelian l-groups with strong unit Olivia Caramello and Anna Carla Russo December 4, 2013 Abstract We show that the theory of MV-algebras is Morita-equivalent
More informationCategorical lattice-valued topology Lecture 1: powerset and topological theories, and their models
Categorical lattice-valued topology Lecture 1: powerset and topological theories, and their models Sergejs Solovjovs Department of Mathematics and Statistics, Faculty of Science, Masaryk University Kotlarska
More informationSUBALGEBRAS AND HOMOMORPHIC IMAGES OF THE RIEGER-NISHIMURA LATTICE
SUBALGEBRAS AND HOMOMORPHIC IMAGES OF THE RIEGER-NISHIMURA LATTICE Guram Bezhanishvili and Revaz Grigolia Abstract In this note we characterize all subalgebras and homomorphic images of the free cyclic
More information8. Distributive Lattices. Every dog must have his day.
8. Distributive Lattices Every dog must have his day. In this chapter and the next we will look at the two most important lattice varieties: distributive and modular lattices. Let us set the context for
More informationCONTINUITY. 1. Continuity 1.1. Preserving limits and colimits. Suppose that F : J C and R: C D are functors. Consider the limit diagrams.
CONTINUITY Abstract. Continuity, tensor products, complete lattices, the Tarski Fixed Point Theorem, existence of adjoints, Freyd s Adjoint Functor Theorem 1. Continuity 1.1. Preserving limits and colimits.
More informationMODAL COMPACT HAUSDORFF SPACES
MODAL COMPACT HAUSDORFF SPACES GURAM BEZHANISHVILI, NICK BEZHANISHVILI, JOHN HARDING Abstract. We introduce modal compact Hausdorff spaces as generalizations of modal spaces, and show these are coalgebras
More informationTopos Theory. Lectures 17-20: The interpretation of logic in categories. Olivia Caramello. Topos Theory. Olivia Caramello.
logic s Lectures 17-20: logic in 2 / 40 logic s Interpreting first-order logic in In Logic, first-order s are a wide class of formal s used for talking about structures of any kind (where the restriction
More informationTones of truth. Andrei Popescu. Technical University Munich and Institute of Mathematics Simion Stoilow of the Romanian Academy
Tones of truth Andrei Popescu Technical University Munich and Institute of Mathematics Simion Stoilow of the Romanian Academy Abstract This paper develops a general algebraic setting for the notion of
More information1 Categories, Functors, and Natural Transformations. Discrete categories. A category is discrete when every arrow is an identity.
MacLane: Categories or Working Mathematician 1 Categories, Functors, and Natural Transormations 1.1 Axioms or Categories 1.2 Categories Discrete categories. A category is discrete when every arrow is an
More informationCoreflections in Algebraic Quantum Logic
Coreflections in Algebraic Quantum Logic Bart Jacobs Jorik Mandemaker Radboud University, Nijmegen, The Netherlands Abstract Various generalizations of Boolean algebras are being studied in algebraic quantum
More informationIII A Functional Approach to General Topology
III A Functional Approach to General Topology Maria Manuel Clementino, Eraldo Giuli and Walter Tholen In this chapter we wish to present a categorical approach to fundamental concepts of General Topology,
More informationEmbedding theorems for classes of GBL-algebras
Embedding theorems for classes of GBL-algebras P. Jipsen and F. Montagna Chapman University, Department of Mathematics and Computer Science, Orange, CA 92866, USA University of Siena, Department of Mathematics
More information10. Finite Lattices and their Congruence Lattices. If memories are all I sing I d rather drive a truck. Ricky Nelson
10. Finite Lattices and their Congruence Lattices If memories are all I sing I d rather drive a truck. Ricky Nelson In this chapter we want to study the structure of finite lattices, and how it is reflected
More informationPROBLEMS, MATH 214A. Affine and quasi-affine varieties
PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset
More informationUniversity of Oxford, Michaelis November 16, Categorical Semantics and Topos Theory Homotopy type theor
Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011 November 16, 2011 References Johnstone, P.T.: Sketches of an Elephant. A Topos-Theory Compendium.
More information``Residuated Structures and Many-valued Logics''
1 Conference on ``Residuated Structures and Many-valued Logics'' Patras, 2-5 June 2004 Abstracts of Invited Speakers 2 FRAMES AND MV-ALGEBRAS L. P. Belluce and A. Di Nola Department of Mathematics British
More informationC -ALGEBRAS MATH SPRING 2015 PROBLEM SET #6
C -ALGEBRAS MATH 113 - SPRING 2015 PROBLEM SET #6 Problem 1 (Positivity in C -algebras). The purpose of this problem is to establish the following result: Theorem. Let A be a unital C -algebra. For a A,
More informationRepresentation of States on MV-algebras by Probabilities on R-generated Boolean Algebras
Representation of States on MV-algebras by Probabilities on R-generated Boolean Algebras Brunella Gerla 1 Tomáš Kroupa 2,3 1. Department of Informatics and Communication, University of Insubria, Via Mazzini
More informationCHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0. Mitya Boyarchenko Vladimir Drinfeld. University of Chicago
CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0 Mitya Boyarchenko Vladimir Drinfeld University of Chicago Some historical comments A geometric approach to representation theory for unipotent
More informationTopos Theory. Lectures 21 and 22: Classifying toposes. Olivia Caramello. Topos Theory. Olivia Caramello. The notion of classifying topos
Lectures 21 and 22: toposes of 2 / 30 Toposes as mathematical universes of Recall that every Grothendieck topos E is an elementary topos. Thus, given the fact that arbitrary colimits exist in E, we can
More informationExtending Algebraic Operations to D-Completions
Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be found at the ENTCS Macro Home Page. Extending Algebraic Operations to D-Completions Klaus
More informationFinite homogeneous and lattice ordered effect algebras
Finite homogeneous and lattice ordered effect algebras Gejza Jenča Department of Mathematics Faculty of Electrical Engineering and Information Technology Slovak Technical University Ilkovičova 3 812 19
More informationCHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0. Mitya Boyarchenko Vladimir Drinfeld. University of Chicago
arxiv:1301.0025v1 [math.rt] 31 Dec 2012 CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0 Mitya Boyarchenko Vladimir Drinfeld University of Chicago Overview These are slides for a talk given
More informationMTL-algebras via rotations of basic hoops
MTL-algebras via rotations of basic hoops Sara Ugolini University of Denver, Department of Mathematics (Ongoing joint work with P. Aglianò) 4th SYSMICS Workshop - September 16th 2018 A commutative, integral
More informationAn Introduction to Thompson s Group V
An Introduction to Thompson s Group V John Fountain 9 March 2011 Properties 1. V contains every finite group 2. V is simple 3. V is finitely presented 4. V has type F P 5. V has solvable word problem 6.
More information