Closure System and Its Semantics
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Arithmetic Equivalence and Isospectrality
ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY ANDREW V.SUTHERLAND ABSTRACT. In these lecture notes we give an introduction to the theory of arithmetic equivalence, a notion originally introduced in a number theoretic setting to refer to number fields with the same zeta function. Gassmann established a direct relationship between arithmetic equivalence and a purely group theoretic notion of equivalence that has since been exploited in several other areas of mathematics, most notably in the spectral theory of Riemannian manifolds by Sunada. We will explicate these results and discuss some applications and generalizations. 1. AN INTRODUCTION TO ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY Let K be a number field (a finite extension of Q), and let OK be its ring of integers (the integral closure of Z in K). The Dedekind zeta function of K is defined by the Dirichlet series X s Y s 1 ζK (s) := N(I)− = (1 N(p)− )− I OK p − ⊆ where the sum ranges over nonzero OK -ideals, the product ranges over nonzero prime ideals, and N(I) := [OK : I] is the absolute norm. For K = Q the Dedekind zeta function ζQ(s) is simply the : P s Riemann zeta function ζ(s) = n 1 n− . As with the Riemann zeta function, the Dirichlet series (and corresponding Euler product) defining≥ the Dedekind zeta function converges absolutely and uniformly to a nonzero holomorphic function on Re(s) > 1, and ζK (s) extends to a meromorphic function on C and satisfies a functional equation, as shown by Hecke [25]. The Dedekind zeta function encodes many features of the number field K: it has a simple pole at s = 1 whose residue is intimately related to several invariants of K, including its class number, and as with the Riemann zeta function, the zeros of ζK (s) are intimately related to the distribution of prime ideals in OK . -
7.2 Binary Operators Closure
last edited April 19, 2016 7.2 Binary Operators A precise discussion of symmetry benefits from the development of what math- ematicians call a group, which is a special kind of set we have not yet explicitly considered. However, before we define a group and explore its properties, we reconsider several familiar sets and some of their most basic features. Over the last several sections, we have considered many di↵erent kinds of sets. We have considered sets of integers (natural numbers, even numbers, odd numbers), sets of rational numbers, sets of vertices, edges, colors, polyhedra and many others. In many of these examples – though certainly not in all of them – we are familiar with rules that tell us how to combine two elements to form another element. For example, if we are dealing with the natural numbers, we might considered the rules of addition, or the rules of multiplication, both of which tell us how to take two elements of N and combine them to give us a (possibly distinct) third element. This motivates the following definition. Definition 26. Given a set S,abinary operator ? is a rule that takes two elements a, b S and manipulates them to give us a third, not necessarily distinct, element2 a?b. Although the term binary operator might be new to us, we are already familiar with many examples. As hinted to earlier, the rule for adding two numbers to give us a third number is a binary operator on the set of integers, or on the set of rational numbers, or on the set of real numbers. -
3. Closed Sets, Closures, and Density
3. Closed sets, closures, and density 1 Motivation Up to this point, all we have done is define what topologies are, define a way of comparing two topologies, define a method for more easily specifying a topology (as a collection of sets generated by a basis), and investigated some simple properties of bases. At this point, we will start introducing some more interesting definitions and phenomena one might encounter in a topological space, starting with the notions of closed sets and closures. Thinking back to some of the motivational concepts from the first lecture, this section will start us on the road to exploring what it means for two sets to be \close" to one another, or what it means for a point to be \close" to a set. We will draw heavily on our intuition about n convergent sequences in R when discussing the basic definitions in this section, and so we begin by recalling that definition from calculus/analysis. 1 n Definition 1.1. A sequence fxngn=1 is said to converge to a point x 2 R if for every > 0 there is a number N 2 N such that xn 2 B(x) for all n > N. 1 Remark 1.2. It is common to refer to the portion of a sequence fxngn=1 after some index 1 N|that is, the sequence fxngn=N+1|as a tail of the sequence. In this language, one would phrase the above definition as \for every > 0 there is a tail of the sequence inside B(x)." n Given what we have established about the topological space Rusual and its standard basis of -balls, we can see that this is equivalent to saying that there is a tail of the sequence inside any open set containing x; this is because the collection of -balls forms a basis for the usual topology, and thus given any open set U containing x there is an such that x 2 B(x) ⊆ U. -
GALOIS THEORY for ARBITRARY FIELD EXTENSIONS Contents 1
GALOIS THEORY FOR ARBITRARY FIELD EXTENSIONS PETE L. CLARK Contents 1. Introduction 1 1.1. Kaplansky's Galois Connection and Correspondence 1 1.2. Three flavors of Galois extensions 2 1.3. Galois theory for algebraic extensions 3 1.4. Transcendental Extensions 3 2. Galois Connections 4 2.1. The basic formalism 4 2.2. Lattice Properties 5 2.3. Examples 6 2.4. Galois Connections Decorticated (Relations) 8 2.5. Indexed Galois Connections 9 3. Galois Theory of Group Actions 11 3.1. Basic Setup 11 3.2. Normality and Stability 11 3.3. The J -topology and the K-topology 12 4. Return to the Galois Correspondence for Field Extensions 15 4.1. The Artinian Perspective 15 4.2. The Index Calculus 17 4.3. Normality and Stability:::and Normality 18 4.4. Finite Galois Extensions 18 4.5. Algebraic Galois Extensions 19 4.6. The J -topology 22 4.7. The K-topology 22 4.8. When K is algebraically closed 22 5. Three Flavors Revisited 24 5.1. Galois Extensions 24 5.2. Dedekind Extensions 26 5.3. Perfectly Galois Extensions 27 6. Notes 28 References 29 Abstract. 1. Introduction 1.1. Kaplansky's Galois Connection and Correspondence. For an arbitrary field extension K=F , define L = L(K=F ) to be the lattice of 1 2 PETE L. CLARK subextensions L of K=F and H = H(K=F ) to be the lattice of all subgroups H of G = Aut(K=F ). Then we have Φ: L!H;L 7! Aut(K=L) and Ψ: H!F;H 7! KH : For L 2 L, we write c(L) := Ψ(Φ(L)) = KAut(K=L): One immediately verifies: L ⊂ L0 =) c(L) ⊂ c(L0);L ⊂ c(L); c(c(L)) = c(L); these properties assert that L 7! c(L) is a closure operator on the lattice L in the sense of order theory. -
Some Order Dualities in Logic, Games and Choices Bernard Monjardet
Some order dualities in logic, games and choices Bernard Monjardet To cite this version: Bernard Monjardet. Some order dualities in logic, games and choices. International Game Theory Review, World Scientific Publishing, 2007, 9 (1), pp.1-12. 10.1142/S0219198907001242. halshs- 00202326 HAL Id: halshs-00202326 https://halshs.archives-ouvertes.fr/halshs-00202326 Submitted on 9 Jan 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 SOME ORDER DUALITIES IN LOGIC, GAMES AND CHOICES B. MONJARDET CERMSEM, Université Paris I, Maison des Sciences Économiques, 106-112 bd de l’Hôpital 75647 Paris Cedex 13, France and CAMS, EHESS. [email protected] 20 February 2004 Abstract We first present the concept of duality appearing in order theory, i.e. the notions of dual isomorphism and of Galois connection. Then we describe two fundamental dualities, the duality extension/intention associated with a binary relation between two sets, and the duality between implicational systems and closure systems. Finally we present two "concrete" dualities occuring in social choice and in choice functions theories. Keywords: antiexchange closure operator, closure system, Galois connection, implicational system, Galois lattice, path-independent choice function, preference aggregation rule, simple game. -
The Poset of Closure Systems on an Infinite Poset: Detachability and Semimodularity Christian Ronse
The poset of closure systems on an infinite poset: Detachability and semimodularity Christian Ronse To cite this version: Christian Ronse. The poset of closure systems on an infinite poset: Detachability and semimodularity. Portugaliae Mathematica, European Mathematical Society Publishing House, 2010, 67 (4), pp.437-452. 10.4171/PM/1872. hal-02882874 HAL Id: hal-02882874 https://hal.archives-ouvertes.fr/hal-02882874 Submitted on 31 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Portugal. Math. (N.S.) Portugaliae Mathematica Vol. xx, Fasc. , 200x, xxx–xxx c European Mathematical Society The poset of closure systems on an infinite poset: detachability and semimodularity Christian Ronse Abstract. Closure operators on a poset can be characterized by the corresponding clo- sure systems. It is known that in a directed complete partial order (DCPO), in particular in any finite poset, the collection of all closure systems is closed under arbitrary inter- section and has a “detachability” or “anti-matroid” property, which implies that the collection of all closure systems is a lower semimodular complete lattice (and dually, the closure operators form an upper semimodular complete lattice). After reviewing the history of the problem, we generalize these results to the case of an infinite poset where closure systems do not necessarily constitute a complete lattice; thus the notions of lower semimodularity and detachability are extended accordingly. -
On Certain Relations for C-Closure Operations on an Ordered Semigroup
International Journal of Pure and Applied Mathematics Volume 92 No. 1 2014, 51-59 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v92i1.4 ijpam.eu ON CERTAIN RELATIONS FOR C-CLOSURE OPERATIONS ON AN ORDERED SEMIGROUP Thawhat Changphas Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen, 40002, THAILAND Abstract: In this paper, a relation for C-closure operations on an ordered semigroup is introduced, using this relation regular and simple ordered semi- groups are characterized. AMS Subject Classification: 06F05 Key Words: semigroup, ordered semigroup, ideal, regular ordered semigroup, simple ordered semigroup, C-closure operation 1. Preliminaries It is known that a semigroup S is regular if and only if it satisfies: A ∩ B = AB for all right ideals A and for all left ideals B of S. Using this property, Pondˇel´iˇcek [2] introduced a relation for C-closure operations on S, and studied some types of semigroups using the relation. The purpose of this paper is to extend Pondˇel´iˇcek’s results to ordered semigroups. In fact, we define a relation for C-closure operations on an ordered semigroup, and characterize regular and simple ordered semigroups using the relation. Firstly, let us recall some certain definitions and results which are in [2]. c 2014 Academic Publications, Ltd. Received: November 7, 2013 url: www.acadpubl.eu 52 T. Changphas Let S be a nonempty set. A mapping U:Su(S) → Su(S) (The symbol Su(S) stands for the set of all subsets of S) is called a C-closure operation on S if, for any A, B in Su(S), it satisfies: (i) U(∅) = ∅; (ii) A ⊆ B ⇒ U(A) ⊆ U(B); (iii) A ⊆ U(A); (iv) U(U(A)) = U(A). -
Math 396. Interior, Closure, and Boundary We Wish to Develop Some Basic Geometric Concepts in Metric Spaces Which Make Precise C
Math 396. Interior, closure, and boundary We wish to develop some basic geometric concepts in metric spaces which make precise certain intuitive ideas centered on the themes of \interior" and \boundary" of a subset of a metric space. One warning must be given. Although there are a number of results proven in this handout, none of it is particularly deep. If you carefully study the proofs (which you should!), then you'll see that none of this requires going much beyond the basic definitions. We will certainly encounter some serious ideas and non-trivial proofs in due course, but at this point the central aim is to acquire some linguistic ability when discussing some basic geometric ideas in a metric space. Thus, the main goal is to familiarize ourselves with some very convenient geometric terminology in terms of which we can discuss more sophisticated ideas later on. 1. Interior and closure Let X be a metric space and A X a subset. We define the interior of A to be the set ⊆ int(A) = a A some Br (a) A; ra > 0 f 2 j a ⊆ g consisting of points for which A is a \neighborhood". We define the closure of A to be the set A = x X x = lim an; with an A for all n n f 2 j !1 2 g consisting of limits of sequences in A. In words, the interior consists of points in A for which all nearby points of X are also in A, whereas the closure allows for \points on the edge of A". -
Arxiv:1301.2793V2 [Math.LO] 18 Jul 2014 Uaeacnetof Concept a Mulate Fpwrbet Uetepwre Xo) N/Rmk S Ff of Use Make And/Or Axiom), Presupp Principles
ON TARSKI’S FIXED POINT THEOREM GIOVANNI CURI To Orsola Abstract. A concept of abstract inductive definition on a complete lattice is formulated and studied. As an application, a constructive version of Tarski’s fixed point theorem is obtained. Introduction The fixed point theorem referred to in this paper is the one asserting that every monotone mapping on a complete lattice L has a least fixed point. The proof, due to A. Tarski, of this result, is a simple and most significant example of a proof that can be carried out on the base of intuitionistic logic (e.g. in the intuitionistic set theory IZF, or in topos logic), and that yet is widely regarded as essentially non-constructive. The reason for this fact is that Tarski’s construction of the fixed point is highly impredicative: if f : L → L is a monotone map, its least fixed point is given by V P , with P ≡ {x ∈ L | f(x) ≤ x}. Impredicativity here is found in the fact that the fixed point, call it p, appears in its own construction (p belongs to P ), and, indirectly, in the fact that the complete lattice L (and, as a consequence, the collection P over which the infimum is taken) is assumed to form a set, an assumption that seems only reasonable in an intuitionistic setting in the presence of strong impredicative principles (cf. Section 2 below). In concrete applications (e.g. in computer science and numerical analysis) the monotone operator f is often also continuous, in particular it preserves suprema of non-empty chains; in this situation, the least fixed point can be constructed taking the supremum of the ascending chain ⊥,f(⊥),f(f(⊥)), ..., given by the set of finite iterations of f on the least element ⊥. -
CLOSURE OPERATORS I Dikran DIKRANJAN Eraldo GIULI
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 27 (1987) 129-143 129 North-Holland CLOSURE OPERATORS I Dikran DIKRANJAN Institute of Mathematics, Bulgarian Academy of Sciences, Sojia, Bulgaria Eraldo GIULI* Department of Pure and Applied Mathematics, University of L’Aquila, 67100-L’Aquila, Italy Received 8 June 1986 Revised 24 September 1986 Closure operators in an (E, M)-category X are introduced as concrete endofunctors of the comma category whose objects are the elements of M. Various kinds of closure operators are studied. There is a Galois equivalence between the conglomerate of idempotent and weakly hereditary closure operators of X and the conglomerate of subclasses of M which are part of a factorization system. There is a one-to-one correspondence between the class of regular closure operators and the class of strongly epireflective subcategories of X. Every closure operator admits an idempotent hull and a weakly hereditary core. Various examples of additive closure operators in Top are given. For abelian categories standard closure operators are considered. It is shown that there is a one-to-one correspondence between the class of standard closure operators and the class of preradicals. Idempotenf weakly hereditary, standard closure operators correspond to idempotent radicals ( = torsion theories). AMS (MOS) Subj. Class.: 18A32, 18B30, 18E40, 54A05, 54B30, 18A40, A-regular morphism weakly hereditary core sequential closure (pre)-radical Introduction Closure operators have always been one of the main concepts in topology. For example, Herrlich [21] characterized coreflections in the category Top of topological spaces by means of Kuratowski closure operators finer than the ordinary closure. -
CLOSURE OPERATORS on ALGEBRAS 3 Extended Power Algebra (P>0A, Ω, ∪) Is a Semilattice Ordered ✵-Algebra
CLOSURE OPERATORS ON ALGEBRAS A. PILITOWSKA1 AND A. ZAMOJSKA-DZIENIO2,∗ Abstract. We study connections between closure operators on an algebra (A, Ω) and congruences on the extended power algebra defined on the same algebra. We use these connections to give an alternative description of the lattice of all subvarieties of semilattice ordered algebras. 1. Introduction In this paper we study closure operators on algebras. M. Novotn´y [12] and J. Fuchs [4] introduced the notion of admissible closure operators for monoids and applied them to study some Galois connections in language theory. In [10] M. Kuˇril and L. Pol´ak used such operators to describe the varieties of semilattice ordered semigroups. In this paper we use a similar approach but apply it to arbitrary alge- bras. In particular, we use concepts from the theory of (extended) power algebras (see e.g. C. Brink [2]). This allows us to generalize and simplify methods presented in [10]. In a natural way one can ”lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. Power algebras of non-empty subsets with the additional operation of set-theoretical union are known as extended power algebras. Extended power algebras play a crucial rˆole in the theory of semilattice ordered algebras. The main aim of this paper is to investigate how congruences on the extended power algebra on (A, Ω) and closure operators defined on (A, Ω) are related. The correspondence obtained allows us to give an alternative (partially simpler) descrip- tion of the subvariety lattice of semilattice ordered algebras, comparing to [16]. -
Completion and Closure Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 41, No 2 (2000), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES DAVID HOLGATE Completion and closure Cahiers de topologie et géométrie différentielle catégoriques, tome 41, no 2 (2000), p. 101-119 <http://www.numdam.org/item?id=CTGDC_2000__41_2_101_0> © Andrée C. Ehresmann et les auteurs, 2000, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation 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/ CAHIERS DE TOPOLOGIE ET Volume XLI-2 (2000) GEOMETRIE DIFFERENTIELLE CATEGORIQUES COMPLETION AND CLOSURE by David HOLGATE RESUME. La fermeture (ou, de fagon synonyme la densite) a tou- jours jou6 un role important dans la th6orie des compl6tions. S’ap- puyant sur des id6es de Birkhoff, une fermeture est extraite de ma- niere canonique d’un processus de completion r6flexive dans une categorie. Cette fermeture caract6rise la compl6tude et la compl6- tion elle-meme. La fermeture n’a pas seulement de bonnes propri6t6s internes, mais c’est la plus grande parmi les fermetures qui d6crivent la completion. Le theoreme principal montre que, equivalent aux descriptions fermeture/densite naturelle d’une completion, est le simple fait mar- quant que les r6flecteurs de completion sont exactement ceux qui pr6servent les plongements. De tels r6flecteurs peuvent 6tre deduits de la fermeture elle-m6me. Le role de la pr6servation de la ferme- ture et du plongement jette alors une nouvelle lumi6re sur les exem- ples de completion.