Notes on Ultrafilters
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Completeness in Quasi-Pseudometric Spaces—A Survey
mathematics Article Completeness in Quasi-Pseudometric Spaces—A Survey ¸StefanCobzas Faculty of Mathematics and Computer Science, Babe¸s-BolyaiUniversity, 400 084 Cluj-Napoca, Romania; [email protected] Received:17 June 2020; Accepted: 24 July 2020; Published: 3 August 2020 Abstract: The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. 17 (1967), 226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., 49 (1984), 36. A discussion on nets defined over ordered or pre-ordered directed sets is also included. Keywords: metric space; quasi-metric space; uniform space; quasi-uniform space; Cauchy sequence; Cauchy net; Cauchy filter; completeness MSC: 54E15; 54E25; 54E50; 46S99 1. Introduction It is well known that completeness is an essential tool in the study of metric spaces, particularly for fixed points results in such spaces. The study of completeness in quasi-metric spaces is considerably more involved, due to the lack of symmetry of the distance—there are several notions of completeness all agreing with the usual one in the metric case (see [1] or [2]). Again these notions are essential in proving fixed point results in quasi-metric spaces as it is shown by some papers on this topic as, for instance, [3–6] (see also the book [7]). -
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. -
Scott Spaces and the Dcpo Category
SCOTT SPACES AND THE DCPO CATEGORY JORDAN BROWN Abstract. Directed-complete partial orders (dcpo’s) arise often in the study of λ-calculus. Here we investigate certain properties of dcpo’s and the Scott spaces they induce. We introduce a new construction which allows for the canonical extension of a partial order to a dcpo and give a proof that the dcpo introduced by Zhao, Xi, and Chen is well-filtered. Contents 1. Introduction 1 2. General Definitions and the Finite Case 2 3. Connectedness of Scott Spaces 5 4. The Categorical Structure of DCPO 6 5. Suprema and the Waybelow Relation 7 6. Hofmann-Mislove Theorem 9 7. Ordinal-Based DCPOs 11 8. Acknowledgments 13 References 13 1. Introduction Directed-complete partially ordered sets (dcpo’s) often arise in the study of λ-calculus. Namely, they are often used to construct models for λ theories. There are several versions of the λ-calculus, all of which attempt to describe the ‘computable’ functions. The first robust descriptions of λ-calculus appeared around the same time as the definition of Turing machines, and Turing’s paper introducing computing machines includes a proof that his computable functions are precisely the λ-definable ones [5] [8]. Though we do not address the λ-calculus directly here, an exposition of certain λ theories and the construction of Scott space models for them can be found in [1]. In these models, computable functions correspond to continuous functions with respect to the Scott topology. It is thus with an eye to the application of topological tools in the study of computability that we investigate the Scott topology. -
Ultrafilter Extensions and the Standard Translation
Notes from Lecture 4: Ultrafilter Extensions and the Standard Translation Eric Pacuit∗ March 1, 2012 1 Ultrafilter Extensions Definition 1.1 (Ultrafilter) Let W be a non-empty set. An ultrafilter on W is a set u ⊆ }(W ) satisfying the following properties: 1. ; 62 u 2. If X; Y 2 u then X \ Y 2 u 3. If X 2 u and X ⊆ Y then Y 2 u. 4. For all X ⊆ W , either X 2 u or X 2 u (where X is the complement of X in W ) / A collection u0 ⊆ }(W ) has the finite intersection property provided for each X; Y 2 u0, X \ Y 6= ;. Theorem 1.2 (Ultrafilter Theorem) If a set u0 ⊆ }(W ) has the finite in- tersection property, then u0 can be extended to an ultrafilter over W (i.e., there is an ultrafilter u over W such that u0 ⊆ u. Proof. Suppose that u0 has the finite intersection property. Then, consider the set u1 = fZ j there are finitely many sets X1;:::Xk such that Z = X1 \···\ Xkg: That is, u1 is the set of finite intersections of sets from u0. Note that u0 ⊆ u1, since u0 has the finite intersection property, we have ; 62 u1, and by definition u1 is closed under finite intersections. Now, define u2 as follows: 0 u = fY j there is a Z 2 u1 such that Z ⊆ Y g ∗UMD, Philosophy. Webpage: ai.stanford.edu/∼epacuit, Email: [email protected] 1 0 0 We claim that u is a consistent filter: Y1;Y2 2 u then there is a Z1 2 u1 such that Z1 ⊆ Y1 and Z2 2 u1 such that Z2 ⊆ Y2. -
Basic Properties of Filter Convergence Spaces
Basic Properties of Filter Convergence Spaces Barbel¨ M. R. Stadlery, Peter F. Stadlery;z;∗ yInstitut fur¨ Theoretische Chemie, Universit¨at Wien, W¨ahringerstraße 17, A-1090 Wien, Austria zThe Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA ∗Address for corresponce Abstract. This technical report summarized facts from the basic theory of filter convergence spaces and gives detailed proofs for them. Many of the results collected here are well known for various types of spaces. We have made no attempt to find the original proofs. 1. Introduction Mathematical notions such as convergence, continuity, and separation are, at textbook level, usually associated with topological spaces. It is possible, however, to introduce them in a much more abstract way, based on axioms for convergence instead of neighborhood. This approach was explored in seminal work by Choquet [4], Hausdorff [12], Katˇetov [14], Kent [16], and others. Here we give a brief introduction to this line of reasoning. While the material is well known to specialists it does not seem to be easily accessible to non-topologists. In some cases we include proofs of elementary facts for two reasons: (i) The most basic facts are quoted without proofs in research papers, and (ii) the proofs may serve as examples to see the rather abstract formalism at work. 2. Sets and Filters Let X be a set, P(X) its power set, and H ⊆ P(X). The we define H∗ = fA ⊆ Xj(X n A) 2= Hg (1) H# = fA ⊆ Xj8Q 2 H : A \ Q =6 ;g The set systems H∗ and H# are called the conjugate and the grill of H, respectively. -
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. -
Certain Ideals Related to the Strong Measure Zero Ideal
数理解析研究所講究録 第 1595 巻 2008 年 37-46 37 Certain ideals related to the strong measure zero ideal 大阪府立大学理学系研究科 大須賀昇 (Noboru Osuga) Graduate school of Science, Osaka Prefecture University 1 Motivation and Basic Definition In 1919, Borel[l] introduced the new class of Lebesgue measure zero sets called strong measure zero sets today. The family of all strong measure zero sets become $\sigma$-ideal and is called the strong measure zero ideal. The four cardinal invariants (the additivity, covering number, uniformity and cofinal- ity) related to the strong measure zero ideal have been studied. In 2002, Yorioka[2] obtained the results about the cofinality of the strong measure zero ideal. In the process, he introduced the ideal $\mathcal{I}_{f}$ for each strictly increas- ing function $f$ on $\omega$ . The ideal $\mathcal{I}_{f}$ relates to the structure of the real line. We are interested in how the cardinal invariants of the ideal $\mathcal{I}_{f}$ behave. $Ma\dot{i}$ly, we te interested in the cardinal invariants of the ideals $\mathcal{I}_{f}$ . In this paper, we deal the consistency problems about the relationship between the cardi- nal invariants of the ideals $\mathcal{I}_{f}$ and the minimam and supremum of cardinal invariants of the ideals $\mathcal{I}_{g}$ for all $g$ . We explain some notation which we use in this paper. Our notation is quite standard. And we refer the reader to [3] and [4] for undefined notation. For sets X and $Y$, we denote by $xY$ the set of all functions $homX$ to Y. -
Basic Topologytaken From
Notes by Tamal K. Dey, OSU 1 Basic Topology taken from [1] 1 Metric space topology We introduce basic notions from point set topology. These notions are prerequisites for more sophisticated topological ideas—manifolds, homeomorphism, and isotopy—introduced later to study algorithms for topological data analysis. To a layman, the word topology evokes visions of “rubber-sheet topology”: the idea that if you bend and stretch a sheet of rubber, it changes shape but always preserves the underlying structure of how it is connected to itself. Homeomorphisms offer a rigorous way to state that an operation preserves the topology of a domain, and isotopy offers a rigorous way to state that the domain can be deformed into a shape without ever colliding with itself. Topology begins with a set T of points—perhaps the points comprising the d-dimensional Euclidean space Rd, or perhaps the points on the surface of a volume such as a coffee mug. We suppose that there is a metric d(p, q) that specifies the scalar distance between every pair of points p, q ∈ T. In the Euclidean space Rd we choose the Euclidean distance. On the surface of the coffee mug, we could choose the Euclidean distance too; alternatively, we could choose the geodesic distance, namely the length of the shortest path from p to q on the mug’s surface. d Let us briefly review the Euclidean metric. We write points in R as p = (p1, p2,..., pd), d where each pi is a real-valued coordinate. The Euclidean inner product of two points p, q ∈ R is d Rd 1/2 d 2 1/2 hp, qi = Pi=1 piqi. -
Model Theory and Ultraproducts
P. I. C. M. – 2018 Rio de Janeiro, Vol. 2 (101–116) MODEL THEORY AND ULTRAPRODUCTS M M Abstract The article motivates recent work on saturation of ultrapowers from a general math- ematical point of view. Introduction In the history of mathematics the idea of the limit has had a remarkable effect on organizing and making visible a certain kind of structure. Its effects can be seen from calculus to extremal graph theory. At the end of the nineteenth century, Cantor introduced infinite cardinals, which allow for a stratification of size in a potentially much deeper way. It is often thought that these ideas, however beautiful, are studied in modern mathematics primarily by set theorists, and that aside from occasional independence results, the hierarchy of infinities has not profoundly influenced, say, algebra, geometry, analysis or topology, and perhaps is not suited to do so. What this conventional wisdom misses is a powerful kind of reflection or transmutation arising from the development of model theory. As the field of model theory has developed over the last century, its methods have allowed for serious interaction with algebraic geom- etry, number theory, and general topology. One of the unique successes of model theoretic classification theory is precisely in allowing for a kind of distillation and focusing of the information gleaned from the opening up of the hierarchy of infinities into definitions and tools for studying specific mathematical structures. By the time they are used, the form of the tools may not reflect this influence. However, if we are interested in advancing further, it may be useful to remember it. -
The Nonstandard Theory of Topological Vector Spaces
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 172, October 1972 THE NONSTANDARDTHEORY OF TOPOLOGICAL VECTOR SPACES BY C. WARD HENSON AND L. C. MOORE, JR. ABSTRACT. In this paper the nonstandard theory of topological vector spaces is developed, with three main objectives: (1) creation of the basic nonstandard concepts and tools; (2) use of these tools to give nonstandard treatments of some major standard theorems ; (3) construction of the nonstandard hull of an arbitrary topological vector space, and the beginning of the study of the class of spaces which tesults. Introduction. Let Ml be a set theoretical structure and let *JR be an enlarge- ment of M. Let (E, 0) be a topological vector space in M. §§1 and 2 of this paper are devoted to the elementary nonstandard theory of (F, 0). In particular, in §1 the concept of 0-finiteness for elements of *E is introduced and the nonstandard hull of (E, 0) (relative to *3R) is defined. §2 introduces the concept of 0-bounded- ness for elements of *E. In §5 the elementary nonstandard theory of locally convex spaces is developed by investigating the mapping in *JK which corresponds to a given pairing. In §§6 and 7 we make use of this theory by providing nonstandard treatments of two aspects of the existing standard theory. In §6, Luxemburg's characterization of the pre-nearstandard elements of *E for a normed space (E, p) is extended to Hausdorff locally convex spaces (E, 8). This characterization is used to prove the theorem of Grothendieck which gives a criterion for the completeness of a Hausdorff locally convex space. -
Topological Spaces and Neighborhood Filters
Topological spaces and neighborhood filters Jordan Bell [email protected] Department of Mathematics, University of Toronto April 3, 2014 If X is a set, a filter on X is a set F of subsets of X such that ; 62 F; if A; B 2 F then A \ B 2 F; if A ⊆ X and there is some B 2 F such that B ⊆ A, then A 2 F. For example, if x 2 X then the set of all subsets of X that include x is a filter on X.1 A basis for the filter F is a subset B ⊆ F such that if A 2 F then there is some B 2 B such that B ⊆ A. If X is a set, a topology on X is a set O of subsets of X such that: ;;X 2 O; S if Uα 2 O for all α 2 I, then Uα 2 O; if I is finite and Uα 2 O for all α 2 I, T α2I then α2I Uα 2 O. If N ⊆ X and x 2 X, we say that N is a neighborhood of x if there is some U 2 O such that x 2 U ⊆ N. In particular, an open set is a neighborhood of every element of itself. A basis for a topology O is a subset B of O such that if x 2 X then there is some B 2 B such that x 2 B, and such that if B1;B2 2 B and x 2 B1 \ B2, then there is some B3 2 B such that 2 x 2 B3 ⊆ B1 \ B2.