First order topological structures and theories

1987 ◽  
Vol 52 (3) ◽  
pp. 763-778 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Topological Structure ◽  
Complex Numbers ◽  
Restricted Class ◽  
Real Numbers ◽  
Boolean Combination ◽  
Topological Structures ◽  
First Order ◽  
Definable Set ◽  
Material Basis ◽  
Definable Sets

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof.Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o-minimal structures [PS] in a general topological context. Note, however, that the p-adic numbers, and structures definable therein, will also fit into our analysis.In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.

Approximate Euler characteristic, dimension, and weak pigeonhole principles

2004 ◽  
Vol 69 (1) ◽  
pp. 201-214
Author(s):  
Jan Krajíček
Keyword(s):  
Euler Characteristic ◽  
Characteristic Dimension ◽  
Order Structure ◽  
Dimension Function ◽  
Pigeonhole Principle ◽  
First Order ◽  
Definable Set ◽  
Definable Sets

AbstractWe define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle : two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardinalities: for any two definable sets A, B either A definably embeds in B or vice versa. Also, a structure admitting a non-trivial approximate Euler characteristic must satisfy .Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle : for no definable set A with more than one element can A2 definably embed into A.

The last word on elimination of quantifiers in modules

1990 ◽  
Vol 55 (2) ◽  
pp. 670-673 ◽  
Author(s):  
Hans B. Gute ◽  
K. K. Reuter
Keyword(s):  
Finite Index ◽  
Finite Family ◽  
Left Translation ◽  
Boolean Combination ◽  
Cartesian Power ◽  
Definable Set ◽  
Combinatorial Lemma ◽  
Definable Subset ◽  
Definable Sets ◽  
The Right

In what follows, a coset is a subset of a group G of the form aH, where H is a subgroup of G; H can be recovered from the coset C: it is the only subgroup which is obtained from C by a left translation; we note in passing that these cosets, that we write systematically with the group to the right, are also of the form Ka, since aH = aHa−1a. A classical combinatorial lemma involving cosets appears in Neumann [1952]: If the coset C = aH is the union of the finite family of cosets C1 = a1H1,…,Cn = anHn, then it is the union of those Ci whose corresponding Hi has finite index in H.In a structure where a group G is defined, Boolean combinations of cosets modulo its definable subgroups form a family of definable sets (by definable, we mean “definable with parameters”). The situation when any definable set is of that kind has been characterized model-theoretically in Hrushovski and Pillay [1987]: A group G is one-based if and only if, for each n, every definable subset of the cartesian power Gn is a(finite!) Boolean combination of cosets modulo definable subgroups. One side is given by a beautiful lemma of Pillay, stating that, in a one-based group which is saturated enough, every type is a right translate of the generic of its left stabilizer.

Some remarks on definable equivalence relations in O-minimal structures

1986 ◽  
Vol 51 (3) ◽  
pp. 709-714 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Equivalence Relation ◽  
Linear Order ◽  
Finite Union ◽  
Equivalence Relations ◽  
Complete Theory ◽  
First Order ◽  
Definable Set ◽  
Order Language ◽  
Minimal Structure ◽  
Definable Sets

Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n-tuples in M, to study definable (in M) equivalence relations on Mn. In particular, we show that if E is an A-definable equivalence relation on Mn (A ⊂ M) then E has only finitely many classes with nonempty interior in Mn, each such class being moreover also A-definable. As a consequence, we are able to give some conditions under which an O-minimal theory T eliminates imaginaries (in the sense of Poizat [P]).If L is a first order language and M an L-structure, then by a definable set in M, we mean something of the form X ⊂ Mn, n ≥ 1, where X = {(a1…,an) ∈ Mn: M ⊨ϕ(ā)} for some formula ∈ L(M). (Here L(M) means L together with names for the elements of M.) If the parameters from come from a subset A of M, we say that X is A-definable.M is said to be O-minimal if M = (M, <,…), where < is a dense linear order with no first or last element, and every definable set X ⊂ M is a finite union of points, and intervals (a, b) (where a, b ∈ M ∪ {± ∞}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory T is said to be O-minimal if every model of T is O-minimal. (Note that in [KPS] it is proved that if M is O-minimal, then T = Th(M) is O-minimal.) In the remainder of this section and in §2, M will denote a fixed but arbitrary O-minimal structure. A,B,C,… will denote subsets of M.

Preliminaries

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Galois Coverings ◽  
Definable Type ◽  
Definable Sets ◽  
Background Material

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

scholarly journals Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

2015 ◽  
Vol 57 (2) ◽  
pp. 157-185 ◽  
Author(s):  
Peter Franek ◽  
Stefan Ratschan ◽  
Piotr Zgliczynski
Keyword(s):  
Order Theory ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

μ-definable sets of integers

1993 ◽  
Vol 58 (1) ◽  
pp. 291-313 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Fixed Point ◽  
Free Variable ◽  
Order Logic ◽  
First Order Logic ◽  
Inductive Definition ◽  
Natural Class ◽  
First Order ◽  
Inductive Definitions ◽  
Definable Sets ◽  
The Universe

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.

scholarly journals Topological Sensitivity in the Recognition of Disoriented Figures

i-Perception ◽  
2018 ◽  
Vol 9 (6) ◽  
pp. 204166951880971 ◽  
Author(s):  
Fumio Kanbe
Keyword(s):  
Structural Properties ◽  
Topological Structure ◽  
Topological Property ◽  
Topological Sensitivity ◽  
Topological Structures ◽  
Invariant Features

A previous study by the author found that discrimination latencies for figure pairs with the same topological structure (isomorphic pairs) were longer than for pairs with different topological structures (nonisomorphic pairs). These results suggest that topological sensitivity occurs during figure recognition. However, sameness was judged in terms of both shape and orientation. Using this criterion, faster discrimination of nonisomorphic pairs may have arisen from the detection of differences in the corresponding locations of the paired figures, which is not a topological property. The current study examined whether topological sensitivity occurs even when identity judgments are based on the sameness of shapes, irrespective of their orientation, where the sameness of location is not ensured. The current results suggested the involvement of topological sensitivity, indicating that processing of structural properties (invariant features) of a figure may be prioritized over processing of superficial features, such as location, length, and angles, in figure recognition.

scholarly journals Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
G. M. Moremedi ◽  
I. P. Stavroulakis
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
First Order ◽  
All Solutions ◽  
Delay Difference ◽  
Constant Argument

Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0,  n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0,  n=0,1,2,…, where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

Encircling Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Nineteenth Century ◽  
Real Number ◽  
Traveling Salesman Problem ◽  
Complex Number ◽  
Traveling Salesman ◽  
Complex Numbers ◽  
Real Numbers ◽  
Hamiltonian Graphs ◽  
The World ◽  
The Traveling Salesman Problem

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

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