Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields
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Directed Sets and Topological Spaces Definable in O-Minimal Structures
Directed sets and topological spaces definable in o-minimal structures. Pablo And´ujarGuerrero∗ Margaret E. M. Thomas ∗ Erik Walsbergy 2010 Mathematics Subject Classification. 03C64 (Primary), 54A20, 54A05, 54D30 (Secondary). Key words. o-minimality, directed sets, definable topological spaces. Abstract We study directed sets definable in o-minimal structures, show- ing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered groups, and furthermore that no analogue holds in full generality. We use the theory of tame pairs to extend the results in the field case to definable families of sets with the finite intersection property. We then apply our results to the study of definable topologies. We prove that all de- finable topological spaces display properties akin to first countability, and give several characterizations of a notion of definable compactness due to Peterzil and Steinhorn [PS99] generalized to this setting. 1 Introduction The study of objects definable in o-minimal structures is motivated by the notion that o-minimality provides a rich but \tame" setting for the theories of said objects. In this paper we study directed sets definable in o-minimal structures, focusing on expansions of groups and fields. By \directed set" we mean a preordered set in which every finite subset has a lower (if downward ∗Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, U.S.A. E-mail addresses: [email protected] (And´ujarGuer- rero), [email protected] (Thomas) yDepartment of Mathematics, Statistics, and Computer Science, Department of Math- ematics, University of California, Irvine, 340 Rowland Hall (Bldg.# 400), Irvine, CA 92697-3875, U.S.A. -
Cofinal Stable Logics
Guram Bezhanishvili Cofinal Stable Logics Nick Bezhanishvili Julia Ilin Abstract. We generalize the (∧, ∨)-canonical formulas to (∧, ∨)-canonical rules, and prove that each intuitionistic multi-conclusion consequence relation is axiomatizable by (∧, ∨)-canonical rules. This yields a convenient characterization of stable superintuition- istic logics. The (∧, ∨)-canonical formulas are analogues of the (∧, →)-canonical formulas, which are the algebraic counterpart of Zakharyaschev’s canonical formulas for superintu- itionistic logics (si-logics for short). Consequently, stable si-logics are analogues of subframe si-logics. We introduce cofinal stable intuitionistic multi-conclusion consequence relations and cofinal stable si-logics, thus answering the question of what the analogues of cofinal subframe logics should be. This is done by utilizing the (∧, ∨, ¬)-reduct of Heyting alge- bras. We prove that every cofinal stable si-logic has the finite model property, and that there are continuum many cofinal stable si-logics that are not stable. We conclude with several examples showing the similarities and differences between the classes of stable, cofinal stable, subframe, and cofinal subframe si-logics. Keywords: Intuitionistic logic, Intuitionistic multi-conclusion consequence relation, Axiomatization, Heyting algebra, Variety, Universal class. Mathematics Subject Classification: 03B55 · 06D20. 1. Introduction Superintuitionistic logics (si-logics for short) are propositional logics extend- ing the intuitionistic propositional calculus IPC. Consistent si-logics are also known as intermediate logics as they are exactly the logics situated between IPC and the classical propositional calculus CPC.BytheG¨odel translation, si-logics are closely related to modal logics extending S4.Ade- tailed account of si-logics and how they relate to modal logics can be found in [8]. -
COFINAL TYPES of ULTRAFILTERS 1. Introduction We Say That a Poset
COFINAL TYPES OF ULTRAFILTERS DILIP RAGHAVAN AND STEVO TODORCEVIC Abstract. We study Tukey types of ultrafilters on !, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafil- ter. The class of basically generated ultrafilters includes all known ultrafilters <! that are not Tukey above [!1] . We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given. 1. Introduction We say that a poset hD; ≤i is directed if any two members of D have an upper bound in D. A set X ⊂ D is unbounded in D if it doesn't have an upper bound in D. A set X ⊂ D is said to be cofinal in D if 8y 2 D9x 2 X [y ≤ x]. Given directed sets D and E, a map f : D ! E is called a Tukey map if the image (under f) of every unbounded subset of D is unbounded in E. A map g : E ! D is called a convergent map if the image (under g) of every cofinal subset of E is cofinal in D. It is easy to see that there is a Tukey map f : D ! E iff there exists a convergent g : E ! D. When this situation obtains, we say that D is Tukey reducible to E, and we write D ≤T E. -
Directed Sets and Cofinal Types by Stevo Todorcevic
transactions of the american mathematical society Volume 290, Number 2, August 1985 DIRECTED SETS AND COFINAL TYPES BY STEVO TODORCEVIC Abstract. We show that 1, w, ax, u x ux and ["iF" are the only cofinal types of directed sets of size S,, but that there exist many cofinal types of directed sets of size continuum. A partially ordered set D is directed if every two elements of D have an upper bound in D. In this note we consider some basic problems concerning directed sets which have their origin in the theory of Moore-Smith convergence in topology [12, 3, 19, 9]. One such problem is to determine "all essential kind of directed sets" needed for defining the closure operator in a given class of spaces [3, p. 47]. Concerning this problem, the following important notion was introduced by J. Tukey [19]. Two directed sets D and E are cofinally similar if there is a partially ordered set C in which both can be embedded as cofinal subsets. He showed that this is an equivalence relation and that D and E are cofinally similar iff there is a convergent map from D into E and also a convergent map from E into D. The equivalence classes of this relation are called cofinal types. This concept has been extensively studied since then by various authors [4, 13, 7, 8]. Already, from the first introduc- tion of this concept, it has been known that 1, w, ccx, w X cox and [w1]<" represent different cofinal types of directed sets of size < Kls but no more than five such types were known. -
Dualities in Equivariant Kasparov Theory
DUALITIES IN EQUIVARIANT KASPAROV THEORY HEATH EMERSON AND RALF MEYER Abstract. We study several duality isomorphisms between equivariant bivari- ant K-theory groups, generalising Kasparov’s first and second Poincar´eduality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised self-maps. The second duality is related to the descrip- ∗ tion of bivariant Kasparov theory for commutative C -algebras by families of elliptic pseudodifferential operators. For many groupoids, both dualities ap- ply to a universal proper G-space. This is a basic requirement for the dual Dirac method and allows us to describe the Baum–Connes assembly map via localisation of categories. Contents 1. Introduction 1 2. Preliminaries on groupoid actions 7 3. EquivariantKasparovtheoryforgroupoids 11 4. The first duality 13 5. Bundles of compact spaces 29 6. The second duality 32 7. Duals for bundles of smooth manifolds 37 8. Conclusion and outlook 52 References 53 1. Introduction The K-homology of a smooth compact manifold M is naturally isomorphic to the (compactly supported) K-theory of its tangent bundle TM via the map that assigns to a K-theory class on TM an elliptic pseudodifferential operator with appropriate symbol. Dually, the K-theory of M is isomorphic to the (locally finite) K-homology of TM. Both statements have bivariant generalisations, which identify Kasparov’s group KK∗ C(M1), C(M2) for two smooth compact manifolds firstly arXiv:0711.0025v3 [math.KT] 24 Jul 2009 ∗ lf C with K (TM1 × M2), secondly with K∗ (TM2 × M1) := KK∗(C0(TM2 × M1), ). In this article, we consider substantial generalisations of these two duality isomor- phisms in which we replace smooth compact manifolds by more general spaces and work equivariantly with respect to actions of locally compact groups or groupoids. -
Coloring Linear Orders with Rado's Partial Order 1
COLORING LINEAR ORDERS WITH RADO'S PARTIAL ORDER RICCARDO CAMERLO AND ALBERTO MARCONE Abstract. Let ¹R be the preorder of embeddability between countable linear orders colored with elements of Rado's partial or- der (a standard example of a wqo which is not a bqo). We show that ¹R has fairly high complexity with respect to Borel reducibil- ity (e.g. if P is a Borel preorder then P ·B ¹R), although its exact classi¯cation remains open. 1. Introduction This note is a contribution to the study of the relations of embed- dability for countable colored linear orderings from the point of view of descriptive set theory. Fixing ! as the support of a countable linear or- dering, we can consider the space LO of all linear orderings on !. This is a standard Borel space (actually a Polish space, see [Kec95]). For our purposes, to color a linear order means assigning to each element of the support an element from a ¯xed countable set C. A colored linear ordering on ! is thus an element of LO £ C!, which is also a standard Borel space. If Q is a partial order on C we have a natural relation of embed- ! 0 0 dability ¹Q on LO £ C de¯ned by letting (v;') ¹Q (v ;' ) if and only if there exists g : ! ! ! such that: (1) 8a; b 2 ! (a v b () g(a) v0 g(b)); (2) 8a 2 !'(a) Q'0(g(a)). 1 Then ¹Q is a §1 preorder which is not Borel and it can be studied in the framework of Borel reducibility. -
Mathematical Logic
Arch. Math. Logic (2009) 48:825–835 DOI 10.1007/s00153-009-0156-0 Mathematical Logic More fine structural global square sequences Martin Zeman Received: 20 February 2009 / Published online: 13 October 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com Abstract We extend the construction of a global square sequence in extender mod- els from Zeman [8] to a construction of coherent non-threadable sequences and give a characterization of stationary reflection at inaccessibles similar to Jensen’s charac- terization in L. Keywords Global quare sequence · Fine structure · Extender model · Weakly compact cardinal · Stationary reflection AMS Subject Classification 03E05 · 03E45 · 03E55 This note presents a fine structural construction of a so-called (κ, A) sequence for certain stationary subsets A of an inaccessible cardinal κ as well as a characterization of weakly compact cardinals in fine structural extender models in terms of station- ary reflection. These results extend analogous results of Jensen for the constructible universe that originate in [3] and are described in more detail in [1]. Although the characterization of weakly compact cardinals in an extender model turns out to be exactly the same as in L, the proof requires a significant amount of extra work. Also, the author believes that the proof presented in this paper is more straightforward than that described in [3] and [1]. The exposition in this paper is based on extender models with Jensen’s λ-indexing of extenders introduced in [4]; see [7] as a reference. The paper builds on previous work on fine structural square sequences in extender models, in particular on [5,6] and Research partially supported by NSF grant DMS-0500799. -
J\'Onsson Posets
JONSSON´ POSETS ROLAND ASSOUS AND MAURICE POUZET To the memory of Bjarni J´onsson Abstract. According to Kearnes and Oman (2013), an ordered set P is J´onsson if it is infinite and the cardinality of every proper initial segment of P is strictly less than the cardinaliy of P . We examine the structure of J´onsson posets. 1. Introduction An ordered set P is J´onsson if it is infinite and the cardinality of every proper initial segment of P is strictly less than the cardinaliy of P . This notion is due to Kearnes and Oman (2013). J´onsson posets, notably the uncountable one, appear in the study of algebras with the J´onnson property (algebras for which proper subalgebras have a cardinality strictly smaller than the algebra). The study of these algebras are the motivation of the paper by Kearnes and Oman [15] where the reader will find detailed information. Countable J´onsson posets occur naturally at the interface of the theory of relations and symbolic dynamic as Theorem 5 and Theorem 4 below illustrate. They were considered in the thesis of the second author [26], without bearing this name, and characterized in [27] under the name of minimal posets. This characterization involves first the notion of well-quasi-order(w.q.o. for short) – a poset P is w.q.o. if it is well founded, that is every non-empty subset A of P has at least a minimal element, and P has no infinite antichain –, next, the decomposition of a well founded poset into levels (for each ordinal α, the α-th level is defined by setting Pα ∶= Min(P ∖ ⋃β<α Pβ) so that P0 is the set Min(P ) of minimal elements of P ; each element x ∈ Pα has height α, denoted by hP (x); the height of P , denoted by h(P ), is the least ordinal α such that arXiv:1712.09442v1 [math.LO] 22 Dec 2017 Pα = ∅) and, finally, the notion of ideal of a poset (every non-empty initial segment which is up-directed). -
Concerning Topological Convergence of Sets
Czechoslovak Mathematical Journal Zdeněk Frolík Concerning topological convergence of sets Czechoslovak Mathematical Journal, Vol. 10 (1960), No. 2, 168–169,170–171,172–180 Persistent URL: http://dml.cz/dmlcz/100401 Terms of use: © Institute of Mathematics AS CR, 1960 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Чехословацкий математический журнал, т. 10 (85) 1960, Прага CONCERNING TOPOLOGICAL CONVERGENCE OF SETS ZDENÈK FROLLK, Praha (Received March 14, 1959) In this paper a convergence of nets of subsets of a topological space is defined. Fundamental properties of this convergence are derived and applied to the set of all points of a connected compact Hausdorff space К at which the space К is not locally connected. In this con nection a generalisation of the theorem of R. L. MOOBE is given (theorem 4.7). 1. TERMINOLOGY AND NOTATION With small modifications the terminology and notation of J. KELLEY [2] is used throughout. For convenience we recall all definitions relating to Moore- Smith convergence. 1.1. A binary relation ^ directs a set A if the set A is non-void, the relation ^ is transitive and reflexive, and for each m and n in A, there exists an element p e A such that both p ^ m and p ^ n. A directed set is a pair (A, ^) such that the relation ^ directs the set A. -
Cofinal Types of Topological Directed Orders 1
COFINAL TYPES OF TOPOLOGICAL DIRECTED ORDERS SLAWOMIR SOLECKI AND STEVO TODORCEVIC Abstract. We investigate the structure of the Tukey ordering among directed orders arising naturally in topology and measure theory. 1. Introduction Let D and E be directed partial orders. A function f : D E is called Tukey if ! the preimage of each bounded subset of E is bounded in D. A function g : E D ! is called convergent if it maps cofinal subsets of E to cofinal subsets of D. The existence of a Tukey map from D to E is equivalent to the existence of a convergent map from E to D. If this relation holds between D and E, we write D T E and ≤ say that D is Tukey reducible to E or D is Tukey below E. We write D T E if ≡ D T E and E T D. The relation D T E is equivalent to the existence of a ≤ ≤ ≡ directed partial order into which both D and E embed as cofinal subsets. We say then that D and E realize the same cofinal type [14]. Tukey ordering was introduced by Tukey [14] in an attempt to illuminate the Moore{Smith theory of net convergence. It was then realized that the relation T ≤ can be useful in calibrating complexities of various directed sets or, more gener- ally, partially ordered sets, see [5], [10], [11], [14]. As demonstrated by Fremlin in [3], Tukey reducibility is central to understanding the results of Bartoszynski and Raisonnier{Stern saying in effect that the σ-ideal of meager subsets of [0; 1] is more additive than the ideal of Lebesgue measure zero subsets of [0; 1]. -
The Category of Cofinal Types. I 387
THE CATEGORYOF COFINAL TYPES. I BY SEYMOUR GINSBURG AND J. R. ISBELL I.1) Introduction. Briefly, this paper introduces a category S£ which we call the category of cofinal types. We construct a concrete representation Sfi of So, and we determine explicitly the part of Sfi corresponding to the cofinal types of bases of open sets in regular topological spaces. These types have a privileged position in^, and we call them canonical types. Succeeding papers in this series will develop further machinery, particularly for the cofinal types of directed sets. The interest of the results depends heavily on the appropriateness of our definitions of convergent functions and equivalence of convergent functions between partially ordered sets (see below, before and after 1.1). In directed sets, a function f: P—>Q is convergent if and only if it takes every cofinal subset to a cofinal subset; in general, P and Q may have branching structure which / must respect. Equivalence of the convergent functions /: P —>Q and g:P—>Q means, except for some trivialities, that every function AC/Ug (i.e., every value h(p) is f{p) or gip)) is convergent. The relation gives us a quotient Sf of the category of all partially ordered sets and convergent functions. Two objects of SS/ become isomorphic in ^if and only if they are cofinally similar. We adapt a construction F from [2] to make several functors F, F°, F* on W such that the quotient category is So. F* induces a dual embedding of So in the category 96* of all complete Boolean algebras and complete homomor- phisms, and a duality between 96* and the full subcategory of canonical types in if. -
Notes on Set Theory
PRELIMINARY TO PCF OMER ZILBERBOIM Abstract. Basic notions for the course on PCF: stationary sets, clubs, club guessing sequences etc. Proofs were taken from the book Set Theory by Jech, Handbook of Set Theory, chapter on cardinal arithmetic by Abraham-Magidor and from Non-Existence of Universal Members (Sh820) by Shelah. Acknowledgments: I wish to thank Monroe Eskew, Petr Glivický, Quinn Culver and Ilan Frank for their useful help in the proofs. 1. Cofinality and König's Theorem Denition 1. 1.For limit ordinals α; β we say that the increasing sequence hαγ : γ < βi is conal in α if limαγ = α. We say that the conality of α is β, if β is the γ!β least limit ordinal such that an increasing β conal sequence exists in α. 2.We can also say that A ⊂ αis conal in α if sup(A) = α. Claim 2. cf(cf(α)) = cf(α). Proof. If cf(α) = β the there exists a sequence hαi : i < βi which is a witness for it. Now suppose that as witnessed by , then cf(β) = γ hβζ : ζ < γi < αβζ : ζ < γ >is conal in α (so if we assumed that cf(β) < β we would have had a contradiction). Claim 3. For every κ , cf(κ) is a cardinal. Proof. If α = cf(κ) is a limit ordinal and not a cardinal, then there is a one to one, onto map f : α ! jαj and we can use this map to dene a jαj sequence, conal in κ: suppose we are givenhκi : i < αi conal in κ, and dene by induction a new sequence :δ0 = κ0, δi+1 = max(δi + 1; κf(i) + 1), for a limit ordinal we dene δi = max([δj + 1; κf(i) + 1) (note that at a limit stage i, the union is less than κ j<i since we assumed cf(κ) = α > jαj ≥ i).