DOCSLIB.ORG

Irreducible Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Irreducible Spaces International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020 Fuzzy *Irreducible Spaces M. Rowthri, B. Amudhambigai in (X,T) such that 1 . That is., cl() 1 in Abstract: In this paper, the concept of * operator on a (X,T) . family of fuzzy open sets in a fuzzy topological space is introduced. Also, the concepts of fuzzy * irreducible Definition 2.6 [7] A fuzzy set in a fuzzy topological space spaces, fuzzy generic sets and fuzzy (X,T) is called fuzzy nowhere dense if there exists no quasi-Sober spaces are initiated and some properties non-zero fuzzy open set in (X,T) such that cl() . are discussed. That is., int(cl()) 0 in (X,T) . Keywords : operator on FO(X,) , fuzzy generic * sets , fuzzy irreducible spaces, fuzzy III. FUZZY IRREDUCIBLE SPACES quasi-Sober spaces Throughout this paper, fuzzy topological space is shortly 2010 AMS Subject Classification: 54A40, 03E72.. denoted by fts. Then FO(X,) , FC(X,) and I. INTRODUCTION X) denote set of all fuzzy open sets, fuzzy closed sets in (X,) and fuzzy points over L.A. Zadeh[8] initiated fuzzy set in 1965. In 1968, Chang [2] characterized fuzzy topological space. Njastad [5] X respectively. introduced open sets. In the same sprit Bin Shahna [1] Definition 3.1 Let (X,) be a fts. A fuzzy operator defined fuzzy open sets and fuzzy closed sets. The *: FO(X,) I X idea of an irreducible or hyperconnected topological space is defined as, if for each FO(X,) with 0 , has been studied by T. Thompson. In this paper, the concept X of operator on a family of fuzzy open sets in a fts is F int() *() and *(0 X ) 0 X . introduced. Also, the concepts of fuzzy irreducible Remark 3.1 It is easy to check that some examples of fuzzy spaces, fuzzy generic sets and fuzzy operators on FO(X,) are the well known fuzzy operators quasi-Sober spaces are initiated and some viz. F int , F int(Fcl) , Fcl(F int) , Fint(Fcl(Fint)) and properties are discussed. Fcl(F int(Fcl)) . Definition 3.2 Let (X,) be a fts and * be a fuzzy II. PRELIMINARIES operator on . Then any FO(X,) is called This section contains basic definitions and preliminary fuzzy * open if *() . Then 1 is called results needed for this paper. X fuzzy closed set. Definition 2.1 [4] Let X be a topological space. * (i) X is irreducible, if X , and whenever Notation 3.1 The family of all fuzzy open (resp. X Z1 Z2 with Zi closed, X Z1 or X Z2 . fuzzy closed) sets in is notated by (ii) Z X is an irreducible component of X if Z is a F *O(X,) (resp. F *C(X,) ). maximal irreducible subset of X. Definition 3.3 For any X in a fts and be a Definition 2.2 [3] A subset F of a topological space is I irreducible if, F A B where A and B then F A or fuzzy operator on , the fuzzy interior of F B . (briefly, F *int() ) is defined by Definition 2.3 [4] A topological space X is said to be F *int() { : and F *O(X,) }. quasi-sober if for every irreducible closed subset has a generic Definition 3.4 For any in a fts and be a point. fuzzy operator on , the fuzzy closure of Definition 2.4 [6] A fuzzy set A is quasi-coincident with (briefly, F *cl() ) is defined by the fuzzy set B iff x X such that A (x) B (x) 1 . Definition 2.5 [7] A fuzzy set in a fuzzy topological space F *cl() { : and F *C(X,) }. (X,T) is called fuzzy dense if there exists no fuzzy closed set Definition 3.5 Any fts is said to be a fuzzy irreducible space, where is a fuzzy operator on if, for any 1, 2 F *O(X,) where and . Revised Manuscript Received on April 15, 2020. 1 0 X , 2 0 X 1q2 * Correspondence Author Definition 3.6 Any I X in M. Rowthri*, Department of Mathematics, Sri Sarada College for Women(Autonomous), Salem, India. Email: [email protected] a fts is said to be fuzzy Dr. B. Amudhambigai, Department of Mathematics, Sri Sarada College irreducible, for Women(Autonomous), Salem, India.. Email: [email protected] Published By: Retrieval Number: F9132038620/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.F9132.059120 208 & Sciences Publication Fuzzy Irreducible Spaces where is a fuzzy operator on if, 0 X and , the only possibility is where , then either Hence . (1 2 ) 1, 2 F *C(X,) * F *cl(). F *C(X,) 1 or 2 . Then the set of all fuzzy Proposition 3.4 Let in fts , irreducible sets is noted by F *I(X,) where be a fuzzy operator on . If Definition 3.7 Any I X is called a fuzzy F *I(X,) , then . maximal irreducible set of fts , where is Proof. Let F *I(X,) . Let be the set of all X a fuzzy operator on , if there is no fuzzy irreducible sets j I such that F *I(X,) such that . Then collection of j , j J where J is an indexed set. Let be the all fuzzy maximal irreducible sets is denoted by set of fuzzy irreducible sets k , k 1, 2,...,l such F *MI(X, ) that 1 2 ... l . Definition 3.8 Let be a fuzzy operator on Let ' k k , k 1, 2,...,l . It is enough to show FO(X1,1) and FO(X 2 , 2 ) in a ftss (X1,1) and that 'F *I(X,) . Let ' 1 2 , where (X 2 , 2 ) respectively. Any function 1, 2 F *C(X,) . Since each f : (X1,1) (X 2 , 2 ) is said to be a fuzzy k F *I(X,) ,i 1, 2,...,l , we get i 1 or continuous function if for every F *O(X , ) , 2 2 Suppose that for some i J , and 1 i 2 . 0 f ()F *O(X1,1) . , then for all Thus Therefore Proposition 3.1 Let be a fuzzy operator on and ' F *MI(X,) . and in a ftss and Proposition 3.5 If is fuzzy irreducible, respectively. Let f : (X , ) (X , ) be a 1 1 2 2 then for any F *O(X,) , F *cl() 1X . bijective and fuzzy continuous function. If Proof. Since is fuzzy irreducible, for any F *I(X , ) then f ()F *I(X , ) 1 1 2 2 two , F *O(X,) where 0 X and 0 X , Proof. Let , F *C(X , ) such that 1 2 2 2 q . Let 1 F *O(X,) such that 1 0 X . If f () ( ) . Then 1 2 1 1X , then F *cl(1) 1X . Assume that 1 1X 1 1 f ( f ()) f (1 2 ) , which implies that and be such that x . Let FO(X,) t1 1 f 1( ) . Then f 1( f ()) , since f is one-one F *O(X,) and 0 such that x . By 1 2 1 1 X t1 1 Thus f 1( ) f 1( ) . Since F *I(X , ) , hypothesis, q . That is, there exist some x where 1 2 1 1 1 1 ti 1 f 1( ) or f 1( ) Thus f () f ( f 1( )) i 1and i J, J is an indexed set such that x with 1 2 1 (X,) * ti 1 ~ xt qxt . Since 1 is an arbitrary fuzzy open set in 1 1 i or f () f ( f (2 )) and so f () 2 . Therefore , it follows that every fuzzy open 1 in . is such* that x i 1 and i J such that ti 1 Remark 3.2 For any in a fts and be a xt 1 with . Therefore is a fuzzy limit point of fuzzy operator on , F *cl() . i . So every fuzzy point over X is a fuzzy limit point of . Proof. Proof is obvious from the definition of fuzzy 1 Thus, every is such that x F *cl() , closure of a fuzzy set . ti Proposition 3.2 If F *I(X,) in fts and F *O(X,) . Hence, fuzzy closure of be a fuzzy operator on , then every fuzzy open set is 1X . F *cl() F *I(X,) . Proof. Assume that and IV. FUZZY QUASI SOBER SPACE F *cl() 1 2 where 1, 2 F *C(X,) . Definition 4.1 Let be a fuzzy operator on X Since F *cl() and F *I(X,) , 1 or in a fts . Let I be a fuzzy . Then F *cl() or F *cl() . X 2 1 2 irreducible closed set. Any fuzzy set I with Hence , is said to be a fuzzy generic set of if Proposition 3.3. If , then F *MI(X,) F *cl() . F *C(X,) . Proof. Let . Then, there is no F *I(X,) such that . By Proposition 3.2, F *cl() F *I(X,) . Since Published By: Retrieval Number: F9132038620/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.F9132.059120 209 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020 Definition 4.2 Let be a fuzzy operator on Proof. Let . Let in a fts . Then is said to be a fuzzy 1, 2 F *C(X 2 , 2 ) such that quasi-Sober space, if for every fuzzy . Since irreducible closed set there exists a fuzzy , . As is injective, . Then generic set. Since is a fuzzy Remark 4.1 Let be a fuzzy operator on in a continuous function, fts . Let Y X and (Y, ) be a fuzzy subspace of Y f 1( ), f 1( ) F *C(X , ) .
Load more

Recommended publications
  • A Zariski Topology for Modules
    A Zariski Topology for Modules∗ Jawad Abuhlail† Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals [email protected] October 18, 2018 Abstract Given a duo module M over an associative (not necessarily commutative) ring R, a Zariski topology is defined on the spectrum Specfp(M) of fully prime R-submodules of M. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration. 1 Introduction The interplay between the properties of a given ring and the topological properties of the Zariski topology defined on its prime spectrum has been studied intensively in the literature (e.g. [LY2006, ST2010, ZTW2006]). On the other hand, many papers considered the so called top modules, i.e. modules whose spectrum of prime submodules attains a Zariski topology (e.g. [Lu1984, Lu1999, MMS1997, MMS1998, Zah2006-a, Zha2006-b]). arXiv:1007.3149v1 [math.RA] 19 Jul 2010 On the other hand, different notions of primeness for modules were introduced and in- vestigated in the literature (e.g. [Dau1978, Wis1996, RRRF-AS2002, RRW2005, Wij2006, Abu2006, WW2009]). In this paper, we consider a notions that was not dealt with, from the topological point of view, so far. Given a duo module M over an associative ring R, we introduce and topologize the spectrum of fully prime R-submodules of M. Motivated by results on the Zariski topology on the spectrum of prime ideals of a commutative ring (e.g. [Bou1998, AM1969]), we investigate the interplay between the topological properties of the obtained space and the module under consideration.
    [Show full text]
  • 1. Introduction Practical Information About the Course. Problem Classes
    1. Introduction Practical information about the course. Problem classes - 1 every 2 weeks, maybe a bit more Coursework – two pieces, each worth 5% Deadlines: 16 February, 14 March Problem sheets and coursework will be available on my web page: http://wwwf.imperial.ac.uk/~morr/2016-7/teaching/alg-geom.html My email address: [email protected] Office hours: Mon 2-3, Thur 4-5 in Huxley 681 Bézout’s theorem. Here is an example of a theorem in algebraic geometry and an outline of a geometric method for proving it which illustrates some of the main themes in algebraic geometry. Theorem 1.1 (Bézout). Let C be a plane algebraic curve {(x, y): f(x, y) = 0} where f is a polynomial of degree m. Let D be a plane algebraic curve {(x, y): g(x, y) = 0} where g is a polynomial of degree n. Suppose that C and D have no component in common (if they had a component in common, then their intersection would obviously be infinite). Then C ∩ D consists of mn points, provided that (i) we work over the complex numbers C; (ii) we work in the projective plane, which consists of the ordinary plane together with some points at infinity (this will be formally defined later in the course); (iii) we count intersections with the correct multiplicities (e.g. if the curves are tangent at a point, it counts as more than one intersection). We will not define intersection multiplicities in this course, but the idea is that multiple intersections resemble multiple roots of a polynomial in one variable.
    [Show full text]
  • Invariant Hypersurfaces 3
    INVARIANT HYPERSURFACES JASON BELL, RAHIM MOOSA, AND ADAM TOPAZ Abstract. The following theorem, which includes as very special cases re- sults of Jouanolou and Hrushovski on algebraic D-varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose φ1,φ2 : Z → X are dominant rational maps from a (possibly nonreduced) irreducible scheme Z of finite-type to an algebraic variety X, with the property that there are infinitely many hyper- surfaces on X whose scheme-theoretic inverse images under φ1 and φ2 agree. Then there is a nonconstant rational function g on X such that gφ1 = gφ2. In the case when Z is also reduced the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou-Hrushovski theorem to generalised algebraic D-varieties and of Cantat’s theorem to self- correspondences. Contents 1. Introduction 2 2. Some differential algebra preliminaries 4 3. The principal algebraic statement 7 4. Proof of the main theorem 12 5. An application to algebraic D-varieties 13 6. Thereducedcaseandanapplicationtorationaldynamics 17 7. Normal varieties equipped with prime divisors 19 8. Positive characteristic 24 References 26 arXiv:1812.08346v2 [math.AG] 5 Jun 2020 Date: June 8, 2020. Keywords: D-variety, dynamical system, foliation, generalised operator, hypersurface, rational map, self-correspondence. 2010 Mathematics Subject Classification. Primary 14E99; Secondary 12H05 and 12H10. Acknowledgements: The authors are grateful for the referee’s careful reading of the original submission, leading to several improvements and corrections.
    [Show full text]
  • Chapter 2 Affine Algebraic Geometry
    Chapter 2 Affine Algebraic Geometry 2.1 The Algebraic-Geometric Dictionary The correspondence between algebra and geometry is closest in affine algebraic geom- etry, where the basic objects are solutions to systems of polynomial equations. For many applications, it suffices to work over the real R, or the complex numbers C. Since important applications such as coding theory or symbolic computation require finite fields, Fq , or the rational numbers, Q, we shall develop algebraic geometry over an arbitrary field, F, and keep in mind the important cases of R and C. For algebraically closed fields, there is an exact and easily motivated correspondence be- tween algebraic and geometric concepts. When the field is not algebraically closed, this correspondence weakens considerably. When that occurs, we will use the case of algebraically closed fields as our guide and base our definitions on algebra. Similarly, the strongest and most elegant results in algebraic geometry hold only for algebraically closed fields. We will invoke the hypothesis that F is algebraically closed to obtain these results, and then discuss what holds for arbitrary fields, par- ticularly the real numbers. Since many important varieties have structures which are independent of the field of definition, we feel this approach is justified—and it keeps our presentation elementary and motivated. Lastly, for the most part it will suffice to let F be R or C; not only are these the most important cases, but they are also the sources of our geometric intuitions. n Let A denote affine n-space over F. This is the set of all n-tuples (t1,...,tn) of elements of F.
    [Show full text]
  • Positivity in Algebraic Geometry I
    Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 48 Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series Bearbeitet von R.K. Lazarsfeld 1. Auflage 2004. Buch. xviii, 387 S. Hardcover ISBN 978 3 540 22533 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1650 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Introduction to Part One Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of pro- jective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated be- havior of linear series defined by line bundles that may not be ample.
    [Show full text]
  • [Li HUISHI] an Introduction to Commutative Algebra(Bookfi.Org)
    An Introduction to Commutative Algebra From the Viewpoint of Normalization This page intentionally left blank From the Viewpoint of Normalization I Jiaying University, China NEW JERSEY * LONDON * SINSAPORE - EElJlNG * SHANGHA' * HONG KONG * TAIPEI CHENNAI Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library AN INTRODUCTION TO COMMUTATIVE ALGEBRA From the Viewpoint of Normalization Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof. may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rasewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-951-2 Printed in Singapore by World Scientific Printers (S)Pte Ltd For Pinpin and Chao This page intentionally left blank Preface Why normalization? Over the years I had been bothered by selecting material for teaching several one-semester courses. When I taught senior undergraduate stu- dents a first course in (algebraic) number theory, or when I taught first-year graduate students an introduction to algebraic geometry, students strongly felt the lack of some preliminaries on commutative algebra; while I taught first-year graduate students a course in commutative algebra by quoting some nontrivial examples from number theory and algebraic geometry, of- ten times I found my students having difficulty understanding.
    [Show full text]
  • 3. Ample and Semiample We Recall Some Very Classical Algebraic Geometry
    3. Ample and Semiample We recall some very classical algebraic geometry. Let D be an in- 0 tegral Weil divisor. Provided h (X; OX (D)) > 0, D defines a rational map: φ = φD : X 99K Y: The simplest way to define this map is as follows. Pick a basis σ1; σ2; : : : ; σm 0 of the vector space H (X; OX (D)). Define a map m−1 φ: X −! P by the rule x −! [σ1(x): σ2(x): ··· : σm(x)]: Note that to make sense of this notation one has to be a little careful. Really the sections don't take values in C, they take values in the fibre Lx of the line bundle L associated to OX (D), which is a 1-dimensional vector space (let us assume for simplicity that D is Carier so that OX (D) is locally free). One can however make local sense of this mor- phism by taking a local trivialisation of the line bundle LjU ' U × C. Now on a different trivialisation one would get different values. But the two trivialisations differ by a scalar multiple and hence give the same point in Pm−1. However a much better way to proceed is as follows. m−1 0 ∗ P ' P(H (X; OX (D)) ): Given a point x 2 X, let 0 Hx = f σ 2 H (X; OX (D)) j σ(x) = 0 g: 0 Then Hx is a hyperplane in H (X; OX (D)), whence a point of 0 ∗ φ(x) = [Hx] 2 P(H (X; OX (D)) ): Note that φ is not defined everywhere.
    [Show full text]
  • Irreducible Components of Varieties of Modules
    IRREDUCIBLE COMPONENTS OF VARIETIES OF MODULES WILLIAM CRAWLEY-BOEVEY AND JAN SCHROER¨ Abstract. We prove some basic results about irreducible components of varieties of modules for an arbitrary finitely generated associative algebra. Our work generalizes results of Kac and Schofield on represen- tations of quivers, but our methods are quite different, being based on deformation theory. 1. Introduction Let k be an algebraically closed field. When studying the representation theory of a finitely generated k-algebra A (associative, with unit), one would often like to classify the finite-dimensional A-modules. In general, however, this is a hopeless—or meaningless—task. For example, the 1-dimensional modules for a f.g. commutative algebra correspond to points of the corre- sponding affine variety, and what should it mean to classify the points? A more basic problem in that case is to compute the irreducible components of the variety. This suggests to study, for each d ≥ 1, the irreducible com- d ponents of the ‘module variety’ modA(k) of d-dimensional A-modules. d d More precisely, modA(k) is the set of A-module structures on k , or equiv- alently the set of k-algebra homomorphisms from A to Md(k). Now a ho- momorphism is determined by its value on a set of generators of A, and this d gives an embedding of modA(k) as a closed subvariety of a suitable product d of copies of Md(k). The group GLd(k) acts by conjugation on modA(k), and the orbits correspond to isomorphism classes of d-dimensional A-modules.
    [Show full text]
  • Irreducible Components of Rigid Spaces Annales De L’Institut Fourier, Tome 49, No 2 (1999), P
    ANNALES DE L’INSTITUT FOURIER BRIAN CONRAD Irreducible components of rigid spaces Annales de l’institut Fourier, tome 49, no 2 (1999), p. 473-541 <http://www.numdam.org/item?id=AIF_1999__49_2_473_0> © Annales de l’institut Fourier, 1999, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 49, 2 (1999), 473-541 IRREDUCIBLE COMPONENTS OF RIGID SPACES by Brian CONRAD Introduction. Let A; be a field complete with respect to a non-trivial non-archime- dean absolute value and let X be a rigid analytic space over k. When X = Sp(A) is affinoid, it is clear what one should mean by the irreducible components of X: the analytic sets Sp(A/p) for the finitely many minimal prime ideals p of the noetherian ring A. However, the globalization of this notion is not as immediate. For schemes and complex analytic spaces, a global theory of irreducible decomposition is well-known ([EGA], Oi, §2, [GAS], 9.2). Some extra commutative algebra is required in order to carry out the basic construction for rigid spaces. In [CM], §1.2, Coleman and Mazur propose an elementary definition of irreducible components for rigid spaces (as well as a related notion that they call a "component part").
    [Show full text]
  • Algebraic Geometry Sample Exam Solutions 2017
    Algebraic Geometry Sample exam solutions 2017 Question 1. Sub-question 1(a). Define an affine algebraic set. (3 marks) An affine algebraic set is a subset V ⊆ An which consists of the common zeros of some finite set of polynomials f1, . , fm with coefficients in k. Alternatively: an affine algebraic set is a set of the form n V = {(x1, . , xn) ∈ A : f1(x1, . , xn) = ··· = fm(x1, . , xn) = 0} for some polynomials f1, . , fm ∈ k[X1,...,Xn]. Sub-question 1(b). Prove that if V is an affine algebraic set, then V is a union of finitely many irreducible closed subsets. You may use Hilbert’s Basis Theorem provided you state it clearly and correctly. (8 marks) Suppose for contradiction that V is an affine algebraic set which cannot be written as a union of finitely many irreducible closed subsets. V is not itself irreducible, so V = V1 ∪ W1 where V1 and W1 are proper closed subsets of V . If each of V1 and W1 is a union of finitely many irreducible closed subsets, then V is also a union of finitely many irreducible closed subsets. So at least one of V1 and W1 is not a union of finitely many irreducible closed subsets. Without loss of generality, we may suppose that V1 is not a union of finitely many irreducible closed subsets. We repeat the argument: V1 is not reducible, so V1 = V2 ∪ W2 where V2 and W2 are proper closed subsets of V . At least one of V2 and W2 is not a union of finitely many irreducible closed subsets; suppose that V2 is not a union of finitely many irreducible closed subsets.
    [Show full text]
  • Spaces of Types in Positive Model Theory
    Spaces of Types in Positive Model Theory Levon Haykazyan November 2, 2018 Abstract We introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first- order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory. Introduction The collection of definable sets is the basic object of study in model theory. Definable sets are the sets of solutions of formulas in a (universal) model. Tarskian model theory focuses on the collection of all first-order formulas, i.e. formulas obtained from atomic one by taking finite conjunctions, fi- nite disjunctions, negation and existential quantification. Under appropriate conditions of stability and its variants, model theoretic methods can be used to bring structure to the collection of definable sets. Oftentimes, however, arXiv:1711.05754v3 [math.LO] 31 Oct 2018 increased iterations of quantification can create more and more complex de- finable sets. And although these complex definable sets may be of little interest themselves, their presence makes model theoretic analysis difficult or impossible. The solution to limit these extra definable sets, offered by the positive model theory, is to only look at positive first-order formulas, i.e. those ob- tained from the atomic ones using finite conjunctions, finite disjunctions and existential quantification (note the absence of the negation). Positive model theory generalises Tarskian model theory in the sense that the latter can be 1 treated in the former by changing the language.
    [Show full text]
  • Arxiv:Math/0509587V6
    THE SPECIALIZATIONS IN A SCHEME FENG-WEN AN Abstract. In this paper we will obtain some further properties for specializations in a scheme. Using these results, we can take a picture for a scheme and a picture for a morphism of schemes. In particular, we will prove that every morphism of schemes is specialization-preserving and of norm not greater than one (under some condition); a necessary and sufficient condition will be given for an injective morphism between irreducible schemes. Introduction Specializations are concrete and intuitive for one to study classical varieties[6]. The results on this topic relating to schemes are mainly presented in Grothendieck’s EGA. In this paper we try to obtain some properties for the specializations in a scheme such as the lengths of specializations. Together with specializations, a scheme can be regarded as a partially ordered set. In §1 we will prove that every morphism of schemes is specialization-preserving (Proposition 1.3 ) and that every specilization in a scheme is contained in an affine open subset (Proposition 1.9 ). Using those results, we can take a picture for a scheme (Remark 1.10 ): A scheme can be described to be a number of trees standing on the ground such that arXiv:math/0509587v6 [math.AG] 5 Jun 2007 each irreducible component is a tree; the generic point of an irreducible component is the root of the cor- responding tree; the closed points of an irreducible component are the top leaves of the corresponding tree; each specialization in an irreducible component are the branches of the corresponding tree.
    [Show full text]