1. Classical Stone Duality
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Stone Coalgebras
Electronic Notes in Theoretical Computer Science 82 No. 1 (2003) URL: http://www.elsevier.nl/locate/entcs/volume82.html 21 pages Stone Coalgebras Clemens Kupke 1 Alexander Kurz 2 Yde Venema 3 Abstract In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor. We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way. This yields a duality between the category of modal algebras and that of coalgebras over the Vietoris functor. Building on this idea, we introduce the notion of a Vietoris polynomial functor over the category of Stone spaces. For each such functor T we establish a link between the category of T -sorted Boolean algebras with operators and the category of Stone coalgebras over T . Applications include a general theorem providing final coalgebras in the category of T -coalgebras. Key words: coalgebra, Stone spaces, Vietoris topology, modal logic, descriptive general frames, Kripke polynomial functors 1 Introduction Technically, every coalgebra is based on a carrier which itself is an object in the so-called base category. Most of the literature on coalgebras either focuses on Set as the base category, or takes a very general perspective, allowing arbitrary base categories, possibly restricted by some constraints. The aim of this paper is to argue that, besides Set, the category Stone of Stone spaces is of relevance as a base category. -
Arxiv:2101.00942V2 [Math.CT] 31 May 2021 01Ascainfrcmuigmachinery
Reiterman’s Theorem on Finite Algebras for a Monad JIŘÍ ADÁMEK∗, Czech Technical University in Prague, Czech Recublic, and Technische Universität Braun- schweig, Germany LIANG-TING CHEN, Academia Sinica, Taiwan STEFAN MILIUS†, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany HENNING URBAT‡, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e. classes of finite algebras closed under finite products, subalgebras and quotients. In this paper, Reiterman’s theorem is generalized to finite Eilenberg- Moore algebras for a monad T on a category D: we prove that a class of finite T-algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monad T associated to the monad T, which gives a categorical view of the construction of the space of profinite terms. b CCS Concepts: • Theory of computation → Algebraic language theory. Additional Key Words and Phrases: Monad, Pseudovariety, Profinite Algebras ACM Reference Format: Jiří Adámek, Liang-Ting Chen, Stefan Milius, and Henning Urbat. 2021. Reiterman’s Theorem on Finite Al- gebras for a Monad. 1, 1 (June 2021), 49 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn 1 INTRODUCTION One of the main principles of both mathematics and computer science is the specification of struc- tures in terms of equational properties. The first systematic study of equations as mathematical objects was pursued by Birkhoff [7] who proved that a class of algebraic structures over a finitary signature Σ can be specified by equations between Σ-terms if and only if it is closed under quo- tient algebras (a.k.a. -
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. -
An Introduction to Nonstandard Analysis 11
AN INTRODUCTION TO NONSTANDARD ANALYSIS ISAAC DAVIS Abstract. In this paper we give an introduction to nonstandard analysis, starting with an ultrapower construction of the hyperreals. We then demon- strate how theorems in standard analysis \transfer over" to nonstandard anal- ysis, and how theorems in standard analysis can be proven using theorems in nonstandard analysis. 1. Introduction For many centuries, early mathematicians and physicists would solve problems by considering infinitesimally small pieces of a shape, or movement along a path by an infinitesimal amount. Archimedes derived the formula for the area of a circle by thinking of a circle as a polygon with infinitely many infinitesimal sides [1]. In particular, the construction of calculus was first motivated by this intuitive notion of infinitesimal change. G.W. Leibniz's derivation of calculus made extensive use of “infinitesimal” numbers, which were both nonzero but small enough to add to any real number without changing it noticeably. Although intuitively clear, infinitesi- mals were ultimately rejected as mathematically unsound, and were replaced with the common -δ method of computing limits and derivatives. However, in 1960 Abraham Robinson developed nonstandard analysis, in which the reals are rigor- ously extended to include infinitesimal numbers and infinite numbers; this new extended field is called the field of hyperreal numbers. The goal was to create a system of analysis that was more intuitively appealing than standard analysis but without losing any of the rigor of standard analysis. In this paper, we will explore the construction and various uses of nonstandard analysis. In section 2 we will introduce the notion of an ultrafilter, which will allow us to do a typical ultrapower construction of the hyperreal numbers. -
1. Introduction in a Topological Space, the Closure Is Characterized by the Limits of the Ultrafilters
Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 08–37 THE ULTRAFILTER CLOSURE IN ZF GONC¸ALO GUTIERRES Abstract: It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF (Zermelo-Fraenkel set theory without the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultrafilter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski Closure operator and u is the Ultrafilter Closure with uX (A) := {x ∈ X : (∃U ultrafilter in X)[U converges to x and A ∈U]}. However, it is possible to built a topological space X for which uX 6= kX , but the open sets are characterized by the ultrafilter convergence. To do so, it is proved that if every set has a free ultrafilter then the Axiom of Countable Choice holds for families of non-empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0-spaces or {R}. Keywords: Ultrafilter Theorem, Ultrafilter Closure. AMS Subject Classification (2000): 03E25, 54A20. 1. Introduction In a topological space, the closure is characterized by the limits of the ultrafilters. Although, in the absence of the Axiom of Choice, this is not a fact anymore. -
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. -
Heyting Duality
Heyting Duality Vikraman Choudhury April, 2018 1 Heyting Duality Pairs of “equivalent” concepts or phenomena are ubiquitous in mathematics, and dualities relate such two different, even opposing concepts. Stone’s Representa- tion theorem (1936), due to Marshall Stone, states the duality between Boolean algebras and topological spaces. It shows that, for classical propositional logic, the Lindenbaum-Tarski algebra of a set of propositions is isomorphic to the clopen subsets of the set of its valuations, thereby exposing an algebraic viewpoint on logic. In this essay, we consider the case for intuitionistic propositional logic, that is, a duality result for Heyting algebras. It borrows heavily from an exposition of Stone and Heyting duality by van Schijndel and Landsman [2017]. 1.1 Preliminaries Definition (Lattice). A lattice is a poset which admits all finite meets and joins. Categorically, it is a (0, 1) − 푐푎푡푒푔표푟푦 (or a thin category) with all finite limits and finite colimits. Alternatively, a lattice is an algebraic structure in the signature (∧, ∨, 0, 1) that satisfies the following axioms. • ∧ and ∨ are each idempotent, commutative, and associative with respective identities 1 and 0. • the absorption laws, 푥 ∨ (푥 ∧ 푦) = 푥, and 푥 ∧ (푥 ∨ 푦) = 푥. Definition (Distributive lattice). A distributive lattice is a lattice in which ∧and ∨ distribute over each other, that is, the following distributivity axioms are satisfied. Categorically, this makes it a distributive category. 1 • 푥 ∨ (푦 ∧ 푧) = (푥 ∨ 푦) ∧ (푥 ∨ 푧) • 푥 ∧ (푦 ∨ 푧) = (푥 ∧ 푦) ∨ (푥 ∧ 푧) Definition (Complements in a lattice). A complement of an element 푥 of a lattice is an element 푦 such that, 푥 ∧ 푦 = 0 and 푥 ∨ 푦 = 1. -
Bitopological Duality for Distributive Lattices and Heyting Algebras Guram Bezhanishvili New Mexico State University
Chapman University Chapman University Digital Commons Engineering Faculty Articles and Research Fowler School of Engineering 2010 Bitopological Duality for Distributive Lattices and Heyting Algebras Guram Bezhanishvili New Mexico State University Nick Bezhanishvili University of Leicester David Gabelaia Razmadze Mathematical Institute Alexander Kurz Chapman University, [email protected] Follow this and additional works at: https://digitalcommons.chapman.edu/engineering_articles Part of the Algebra Commons, Logic and Foundations Commons, Other Computer Engineering Commons, Other Computer Sciences Commons, and the Other Mathematics Commons Recommended Citation G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, and A. Kurz, “Bitopological duality for distributive lattices and Heyting algebras,” Mathematical Structures in Computer Science, vol. 20, no. 03, pp. 359–393, Jun. 2010. DOI: 10.1017/S0960129509990302 This Article is brought to you for free and open access by the Fowler School of Engineering at Chapman University Digital Commons. It has been accepted for inclusion in Engineering Faculty Articles and Research by an authorized administrator of Chapman University Digital Commons. For more information, please contact [email protected]. Bitopological Duality for Distributive Lattices and Heyting Algebras Comments This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Mathematical Structures in Computer Science, volume 20, number 3, in 2010 following peer review. The definitive publisher- authenticated -
An Introduction to Nonstandard Analysis
An Introduction to Nonstandard Analysis Jeffrey Zhang Directed Reading Program Fall 2018 December 5, 2018 Motivation • Developing/Understanding Differential and Integral Calculus using infinitely large and small numbers • Provide easier and more intuitive proofs of results in analysis Filters Definition Let I be a nonempty set. A filter on I is a nonempty collection F ⊆ P(I ) of subsets of I such that: • If A; B 2 F , then A \ B 2 F . • If A 2 F and A ⊆ B ⊆ I , then B 2 F . F is proper if ; 2= F . Definition An ultrafilter is a proper filter such that for any A ⊆ I , either A 2 F or Ac 2 F . F i = fA ⊆ I : i 2 Ag is called the principal ultrafilter generated by i. Filters Theorem Any infinite set has a nonprincipal ultrafilter on it. Pf: Zorn's Lemma/Axiom of Choice. The Hyperreals Let RN be the set of all real sequences on N, and let F be a fixed nonprincipal ultrafilter on N. Define an (equivalence) relation on RN as follows: hrni ≡ hsni iff fn 2 N : rn = sng 2 F . One can check that this is indeed an equivalence relation. We denote the equivalence class of a sequence r 2 RN under ≡ by [r]. Then ∗ R = f[r]: r 2 RNg: Also, we define [r] + [s] = [hrn + sni] [r] ∗ [s] = [hrn ∗ sni] The Hyperreals We say [r] = [s] iff fn 2 N : rn = sng 2 F . < is defined similarly. ∗ ∗ A subset A of R can be enlarged to a subset A of R, where ∗ [r] 2 A () fn 2 N : rn 2 Ag 2 F : ∗ ∗ ∗ Likewise, a function f : R ! R can be extended to f : R ! R, where ∗ f ([r]) := [hf (r1); f (r2); :::i] The Hyperreals A hyperreal b is called: • limited iff jbj < n for some n 2 N. -
REVERSIBLE FILTERS 1. Introduction a Topological Space X Is Reversible
REVERSIBLE FILTERS ALAN DOW AND RODRIGO HERNANDEZ-GUTI´ ERREZ´ Abstract. A space is reversible if every continuous bijection of the space onto itself is a homeomorphism. In this paper we study the question of which countable spaces with a unique non-isolated point are reversible. By Stone duality, these spaces correspond to closed subsets in the Cech-Stoneˇ compact- ification of the natural numbers β!. From this, the following natural problem arises: given a space X that is embeddable in β!, is it possible to embed X in such a way that the associated filter of neighborhoods defines a reversible (or non-reversible) space? We give the solution to this problem in some cases. It is especially interesting whether the image of the required embedding is a weak P -set. 1. Introduction A topological space X is reversible if every time that f : X ! X is a continuous bijection, then f is a homeomorphism. This class of spaces was defined in [10], where some examples of reversible spaces were given. These include compact spaces, Euclidean spaces Rn (by the Brouwer invariance of domain theorem) and the space ! [ fpg, where p is an ultrafilter, as a subset of β!. This last example is of interest to us. Given a filter F ⊂ P(!), consider the space ξ(F) = ! [ fFg, where every point of ! is isolated and every neighborhood of F is of the form fFg [ A with A 2 F. Spaces of the form ξ(F) have been studied before, for example by Garc´ıa-Ferreira and Uzc´ategi([6] and [7]). -
Some Limits of Boolean Algebras
SOME LIMITS OF BOOLEAN ALGEBRAS FRANKLIN HAIMO 1. Introduction. Inverse systems of topological spaces and direct and inverse systems of Abelian groups and resulting applications thereof to algebraic topology have been studied intensively, and the techniques employed are now standard [2]. (Numbers in brackets refer to the bibliography at the end of the paper.) We shall apply some of these ideas to the treatment of sets of Boolean algebras in order to obtain, in the inverse case, a representation, and, in both cases, some information about the ideals and about the decompositions of certain algebras. For instance, it is well known [l] that the cardinal product B* of a set {Ba} of Boolean algebras, indexed by a class A, is also a Boolean algebra. Moreover, if each Ba is m-complete for some fixed cardinal m [3], then B* is m-complete. When A is a set partly ordered by = and directed thereby, and when the Ba form an inverse system (to be defined below), we shall show that the re- sulting inverse limit set is a Boolean algebra, a subalgebra of B*, and a similar result will hold in the direct case, at least in part.1 For each nonfinite Boolean algebra B, there will be constructed a nontrivial inverse limit representation of B, the inverse limit of all principal ideals [3] of B (with repetitions). In the direct case, ideals in the limit algebra (and the homomorphic algebras which they generate) will turn out to be direct limits of like objects. Direct decomposition into ideals of the direct limit takes place (it will be seen) if there is a similar decomposition in each Ba. -
SELECTIVE ULTRAFILTERS and Ω −→ (Ω) 1. Introduction Ramsey's
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 10, Pages 3067{3071 S 0002-9939(99)04835-2 Article electronically published on April 23, 1999 SELECTIVE ULTRAFILTERS AND ! ( ! ) ! −→ TODD EISWORTH (Communicated by Andreas R. Blass) Abstract. Mathias (Happy families, Ann. Math. Logic. 12 (1977), 59{ 111) proved that, assuming the existence of a Mahlo cardinal, it is consistent that CH holds and every set of reals in L(R)is -Ramsey with respect to every selective ultrafilter . In this paper, we showU that the large cardinal assumption cannot be weakened.U 1. Introduction Ramsey's theorem [5] states that for any n !,iftheset[!]n of n-element sets of natural numbers is partitioned into finitely many∈ pieces, then there is an infinite set H ! such that [H]n is contained in a single piece of the partition. The set H is said⊆ to be homogeneous for the partition. We can attempt to generalize Ramsey's theorem to [!]!, the collection of infinite subsets of the natural numbers. Let ! ( ! ) ! abbreviate the statement \for every [!]! there is an infinite H ! such−→ that either [H]! or [H]! = ." UsingX⊆ the axiom of choice, it is easy⊆ to partition [!]! into two⊆X pieces in such∩X a way∅ that any two infinite subsets of ! differing by a single element lie in different pieces of the partition. Obviously such a partition can admit no infinite homogeneous set. Thus under the axiom of choice, ! ( ! ) ! is false. Having been stymied in this naive−→ attempt at generalization, we have several obvious ways to proceed. A natural project would be to attempt to find classes of partitions of [!]! that do admit infinite homogeneous sets.