Order Types and Structure of Orders
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Handout from Today's Lecture
MA532 Lecture Timothy Kohl Boston University April 23, 2020 Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 1 / 26 Cardinal Arithmetic Recall that one may define addition and multiplication of ordinals α = ot(A, A) β = ot(B, B ) α + β and α · β by constructing order relations on A ∪ B and B × A. For cardinal numbers the foundations are somewhat similar, but also somewhat simpler since one need not refer to orderings. Definition For sets A, B where |A| = α and |B| = β then α + β = |(A × {0}) ∪ (B × {1})|. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 2 / 26 The curious part of the definition is the two sets A × {0} and B × {1} which can be viewed as subsets of the direct product (A ∪ B) × {0, 1} which basically allows us to add |A| and |B|, in particular since, in the usual formula for the size of the union of two sets |A ∪ B| = |A| + |B| − |A ∩ B| which in this case is bypassed since, by construction, (A × {0}) ∩ (B × {1})= ∅ regardless of the nature of A ∩ B. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 3 / 26 Definition For sets A, B where |A| = α and |B| = β then α · β = |A × B|. One immediate consequence of these definitions is the following. Proposition If m, n are finite ordinals, then as cardinals one has |m| + |n| = |m + n|, (where the addition on the right is ordinal addition in ω) meaning that ordinal addition and cardinal addition agree. Proof. The simplest proof of this is to define a bijection f : (m × {0}) ∪ (n × {1}) → m + n by f (hr, 0i)= r for r ∈ m and f (hs, 1i)= m + s for s ∈ n. -
Types for Describing Coordinated Data Structures
Types for Describing Coordinated Data Structures Michael F. Ringenburg∗ Dan Grossman [email protected] [email protected] Dept. of Computer Science & Engineering University of Washington, Seattle, WA 98195 ABSTRACT the n-th function). This section motivates why such invari- Coordinated data structures are sets of (perhaps unbounded) ants are important, explores why the scope of type variables data structures where the nodes of each structure may share makes the problem appear daunting, and previews the rest abstract types with the corresponding nodes of the other of the paper. structures. For example, consider a list of arguments, and 1.1 Low-Level Type Systems a separate list of functions, where the n-th function of the Recent years have witnessed substantial work on powerful second list should be applied only to the n-th argument of type systems for safe, low-level languages. Standard moti- the first list. We can guarantee that this invariant is obeyed vation for such systems includes compiler debugging (gener- by coordinating the two lists, such that the type of the n-th ated code that does not type check implies a compiler error), argument is existentially quantified and identical to the ar- proof-carrying code (the type system encodes a safety prop- gument type of the n-th function. In this paper, we describe erty that the type-checker verifies), automated optimization a minimal set of features sufficient for a type system to sup- (an optimizer can exploit the type information), and manual port coordinated data structures. We also demonstrate that optimization (humans can use idioms unavailable in higher- two known type systems (Crary and Weirich’s LX [6] and level languages without sacrificing safety). -
Multiplication in Vector Lattices
MULTIPLICATION IN VECTOR LATTICES NORMAN M. RICE 1. Introduction. B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)). In general, not every pair of elements in L will have a product in L. In § 3 we discuss certain properties which ensure that, in fact, the multiplication will be universally defined, and it turns out that L can always be embedded as an order-dense order ideal in a larger space L# which has these properties. It is then possible to define multiplication in spaces without a unit. In § 4 we show that if L has a normal integral 0, then 4> can be extended to a normal integral on a larger space Li(<t>) in L#, and £i(0) may be regarded as an abstract integral space. In § 5 a very general form of the Radon-Niko- dym theorem is proved, and in § 6 this is used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition for the Radon-Nikodym theorem to hold in a measure space. 2. Multiplication in spaces with a unit. Let L be a vector lattice which is Dedekind complete (i.e., every set which is bounded above has a least upper bound) and has a weak order unit 1 (i.e., inf (1, x) > 0, whenever x > 0). -
Dimensional Functions Over Partially Ordered Sets
Intro Dimensional functions over posets Application I Application II Dimensional functions over partially ordered sets V.N.Remeslennikov, E. Frenkel May 30, 2013 1 / 38 Intro Dimensional functions over posets Application I Application II Plan The notion of a dimensional function over a partially ordered set was introduced by V. N. Remeslennikov in 2012. Outline of the talk: Part I. Definition and fundamental results on dimensional functions, (based on the paper of V. N. Remeslennikov and A. N. Rybalov “Dimensional functions over posets”); Part II. 1st application: Definition of dimension for arbitrary algebraic systems; Part III. 2nd application: Definition of dimension for regular subsets of free groups (L. Frenkel and V. N. Remeslennikov “Dimensional functions for regular subsets of free groups”, work in progress). 2 / 38 Intro Dimensional functions over posets Application I Application II Partially ordered sets Definition A partial order is a binary relation ≤ over a set M such that ∀a ∈ M a ≤ a (reflexivity); ∀a, b ∈ M a ≤ b and b ≤ a implies a = b (antisymmetry); ∀a, b, c ∈ M a ≤ b and b ≤ c implies a ≤ c (transitivity). Definition A set M with a partial order is called a partially ordered set (poset). 3 / 38 Intro Dimensional functions over posets Application I Application II Linearly ordered abelian groups Definition A set A equipped with addition + and a linear order ≤ is called linearly ordered abelian group if 1. A, + is an abelian group; 2. A, ≤ is a linearly ordered set; 3. ∀a, b, c ∈ A a ≤ b implies a + c ≤ b + c. Definition The semigroup A+ of all nonnegative elements of A is defined by A+ = {a ∈ A | 0 ≤ a}. -
Generic Haskell: Applications
Generic Haskell: applications Ralf Hinze1 and Johan Jeuring2,3 1 Institut f¨urInformatik III, Universit¨atBonn R¨omerstraße 164, 53117 Bonn, Germany [email protected] http://www.informatik.uni-bonn.de/~ralf/ 2 Institute of Information and Computing Sciences, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, The Netherlands [email protected] http://www.cs.uu.nl/~johanj/ 3 Open University, Heerlen, The Netherlands Abstract. 1 Generic Haskell is an extension of Haskell that supports the construc- tion of generic programs. This article describes generic programming in practice. It discusses three advanced generic programming applications: generic dictionaries, compressing XML documents, and the zipper. When describing and implementing these examples, we will encounter some advanced features of Generic Haskell, such as type-indexed data types, dependencies between and generic abstractions of generic functions, ad- justing a generic function using a default case, and generic functions with a special case for a particular constructor. 1 Introduction A polytypic (or generic, type-indexed) function is a function that can be instan- tiated on many data types to obtain data type specific functionality. Examples of polytypic functions are the functions that can be derived in Haskell [50], such as show, read, and ‘ ’. In [23] we have introduced type-indexed functions, and we have shown how to implement them in Generic Haskell [7]. For an older introduction to generic programming, see Backhouse et al [4]. Why is generic programming important? Generic programming makes pro- grams easier to write: – Programs that could only be written in an untyped style can now be written in a language with types. -
Mouse Pairs and Suslin Cardinals
Mouse pairs and Suslin cardinals John R. Steel October 2018 Abstract Building on the results of [13] and [12], we obtain optimal Suslin representations for M mouse pairs. This leads us to an analysis of HOD , for any model M of ADR in which there are Wadge-cofinally many mouse pairs. We show also that if there is a least branch hod pair with a Woodin limit of Woodin cardinals, then there is a model of AD+ in which not every set of reals is ordinal definable from a countable sequence of ordinals. 0 Introduction The paper [13] introduces mouse pairs, and develops their basic theory. It then uses one variety of mouse pair, the least branch hod pairs, to analyze HOD in certain models of the Axiom of Determinacy. Here we shall carry that work further, by obtaining optimal Suslin representations for mouse pairs. This enables us to show to show that the collection of models M of ADR to which the HOD-analysis of [13] applies is closed downward under Wadge reducibility. It also leads to a characterization of the Solovay sequence in such models M of ADR in terms of the Woodin cardinals of HODM . Unless otherwise stated, we shall assume AD+ throughout this paper. 0.1 Mouse pairs Our intended reader has already read [13], but let us recall, in outline, some of the definitions and results of that paper. _ E~ ~ ~ A pure extender premouse is a transitive structure M = (Jα ; 2; E; F ), where E F is a M coherent sequence of extenders. (F = ; is allowed.) We use Jensen indexing: if E = Eα , then +;M α = iE(crit(E) ). -
Omega-Models of Finite Set Theory
ω-MODELS OF FINITE SET THEORY ALI ENAYAT, JAMES H. SCHMERL, AND ALBERT VISSER Abstract. Finite set theory, here denoted ZFfin, is the theory ob- tained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An ω-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) em- ployed the Bernays-Rieger method of permutations to construct a recursive ω-model of ZFfin that is nonstandard (i.e., not isomor- phic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ω-models of ZFfin. In par- ticular, we present a new method for building ω-models of ZFfin that leads to a perspicuous construction of recursive nonstandard ω-models of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an ω-model. The central theorem of the paper is the following: Theorem A. For every graph (A, F ), where F is a set of un- ordered pairs of A, there is an ω-model M of ZFfin whose universe contains A and which satisfies the following conditions: (1) (A, F ) is definable in M; (2) Every element of M is definable in (M, a)a∈A; (3) If (A, F ) is pointwise definable, then so is M; (4) Aut(M) =∼ Aut(A, F ). Theorem A enables us to build a variety of ω-models with special features, in particular: Corollary 1. Every group can be realized as the automorphism group of an ω-model of ZFfin. -
Chapter 8 Ordered Sets
Chapter VIII Ordered Sets, Ordinals and Transfinite Methods 1. Introduction In this chapter, we will look at certain kinds of ordered sets. If a set \ is ordered in a reasonable way, then there is a natural way to define an “order topology” on \. Most interesting (for our purposes) will be ordered sets that satisfy a very strong ordering condition: that every nonempty subset contains a smallest element. Such sets are called well-ordered. The most familiar example of a well-ordered set is and it is the well-ordering property that lets us do mathematical induction in In this chapter we will see “longer” well ordered sets and these will give us a new proof method called “transfinite induction.” But we begin with something simpler. 2. Partially Ordered Sets Recall that a relation V\ on a set is a subset of \‚\ (see Definition I.5.2 ). If ÐBßCÑ−V, we write BVCÞ An “order” on a set \ is refers to a relation on \ that satisfies some additional conditions. Order relations are usually denoted by symbols such asŸ¡ß£ , , or . Definition 2.1 A relation V\ on is called: transitive ifÀ a +ß ,ß - − \ Ð+V, and ,V-Ñ Ê +V-Þ reflexive ifÀa+−\+V+ antisymmetric ifÀ a +ß , − \ Ð+V, and ,V+ Ñ Ê Ð+ œ ,Ñ symmetric ifÀ a +ß , − \ +V, Í ,V+ (that is, the set V is “symmetric” with respect to thediagonal ? œÖÐBßBÑÀB−\ש\‚\). Example 2.2 1) The relation “œ\ ” on a set is transitive, reflexive, symmetric, and antisymmetric. Viewed as a subset of \‚\, the relation “ œ ” is the diagonal set ? œÖÐBßBÑÀB−\×Þ 2) In ‘, the usual order relation is transitive and antisymmetric, but not reflexive or symmetric. -
When Is a Lots Densely Orderable?
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 30 (1988) 225-235 225 North-Holland WHEN IS A LOTS DENSELY ORDERABLE? Victor NEUMANN-LARA Institute de Matetmiticas, Uniuersidad National Autdnoma de M&ico, Ciudad Universitaria, Mhico 20, D.F. Richard C. WILSON Departamento de Matemdticas, Universidad Autdnoma Metropolitana, Unidad Iztapalapa, M&ico 13, D.F. Received 15 May 1986 Revised 19 October 1987 We obtain sufficient conditions for a linearly ordered topological space (LOTS) to be densely orderable. As a consequence we obtain a characterization of those metrizable LOTS with connected ordered compactifications. AMS (MOS) Subj. Class.: Primary 54F05 1. Introduction A well-known result states that the weight w of a linearly ordered topological space (LOTS) is equal to the sum of the density d and the number of jumps j. Thus if j > d, then w =j and hence j is an order-topological invariant of the space in the sense that any other order which induces the same topology will have the same number of jumps. On the other hand, if j G d (in particular if j = Ko) then this is not necessarily the case-for example, the space of rationals Q with the order topology is homeomophic to Q x (0, l} with the lexicographic order topology. In this paper we will obtain conditions under which a LOTS can be reordered without jumps, that is to say under which there exists a dense order inducing the same topology. 2. -
Order on Order Types∗
Order on Order Types∗ Alexander Pilz1 and Emo Welzl2 1 Institute for Software Technology, Graz University of Technology, Austria [email protected] 2 Department of Computer Science, Institute of Theoretical Computer Science ETH Zürich, Switzerland [email protected] Abstract Given P and P 0, equally sized planar point sets in general position, we call a bijection from P to P 0 crossing-preserving if crossings of connecting segments in P are preserved in P 0 (extra crossings may occur in P 0). If such a mapping exists, we say that P 0 crossing-dominates P , and if such a mapping exists in both directions, P and P 0 are called crossing-equivalent. The relation is transitive, and we have a partial order on the obtained equivalence classes (called crossing types or x-types). Point sets of equal order type are clearly crossing-equivalent, but not vice versa. Thus, x-types are a coarser classification than order types. (We will see, though, that a collapse of different order types to one x-type occurs for sets with triangular convex hull only.) We argue that either the maximal or the minimal x-types are sufficient for answering many combinatorial (existential or extremal) questions on planar point sets. Motivated by this we consider basic properties of the relation. We characterize order types crossing-dominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomial-time algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points. -
General-Well-Ordered Sets
J. Austral. Math. Soc. 19 (Series A), (1975), 7-20. GENERAL-WELL-ORDERED SETS J. L. HICKMAN (Received 10 April 1972) Communicated by J. N. Crossley It is of course well known that within the framework of any reasonable set theory whose axioms include that of choice, we can characterize well-orderings in two different ways: (1) a total order for which every nonempty subset has a minimal element; (2) a total order in which there are no infinite descending chains. Now the theory of well-ordered sets and their ordinals that is expounded in various texts takes as its definition characterization (1) above; in this paper we commence an investigation into the corresponding theory that takes character- ization (2) as its starting point. Naturally if we are to obtain any differences at all, we must exclude the axiom of choice from our set theory. Thus we state right at the outset that we are working in Zermelo-Fraenkel set theory without choice. This is not enough, however, for it may turn out that the axiom of choice is not really necessary for the equivalence of (1) and (2), and if this were so, we would still not obtain any deviation from the "classical" theory. Thus the first section of this paper will be devoted to showing that we can in fact have sets that are well-ordered according to (2) but not according to (1). Naturally we have no hope of reversing this situation, for it is a ZF-theorem that any set well-ordered by (1) is also well-ordered by (2). -
Logic and Rational Languages of Words Indexed by Linear Orderings
Logic and Rational Languages of Words Indexed by Linear Orderings Nicolas Bedon1, Alexis B`es2, Olivier Carton3, and Chlo´eRispal1 1 Universit´eParis-Est and CNRS Laboratoire d’informatique de l’Institut Gaspard Monge, CNRS UMR 8049 Email: [email protected], [email protected] 2 Universit´eParis-Est, LACL Email: [email protected], 3 Universit´eParis 7 and CNRS LIAFA, CNRS UMR 7089 Email: [email protected] Abstract. We prove that every rational language of words indexed by linear orderings is definable in monadic second-order logic. We also show that the converse is true for the class of languages indexed by countable scattered linear orderings, but false in the general case. As a corollary we prove that the inclusion problem for rational languages of words indexed by countable linear orderings is decidable. 1 Introduction In [4, 6], Bruy`ere and Carton introduce automata and rational expressions for words on linear orderings. These notions unify naturally previously defined no- tions for finite words, left- and right-infinite words, bi-infinite words, and ordinal words. They also prove that a Kleene-like theorem holds when the orderings are restricted to countable scattered linear orderings; recall that a linear ordering is scattered if it does not contain any dense sub-ordering. Since [4], the study of automata on linear orderings was carried on in several papers. The emptiness problem and the inclusion problem for rational languages is addressed in [7, 11]. The papers [5, 2] provide a classification of rational languages with respect to the rational operations needed to describe them.