Expansion of a model of a weakly o-minimal theory by a family of unary predicates

2001 ◽  
Vol 66 (3) ◽  
pp. 1382-1414 ◽  
Author(s):  
Bektur Sembiuly Baizhanov
Keyword(s):  
Finite Number ◽  
Convex Sets ◽  
Finite Union ◽  
Complete Theory ◽  
Ordered Structure ◽  
First Order ◽  
Minimal Theory

AbstractA subset A ⊆ M of a totally ordered structure M is said to be convex, if for any a, b ∈ A: [a < b → ∀t (a < tb → t ∈ A)]. A complete theory of first order is weakly o-minimal (M. Dickmann [D]) if any model M is totally ordered by some ∅-definable formula and any subset of M which is definable with parameters from M is a finite union of convex sets. We prove here that for any model M of a weakly o-minimal theory T. any expansion M+ of M by a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63). that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories.

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.

Omitting types in -minimal theories

1986 ◽  
Vol 51 (1) ◽  
pp. 63-74 ◽  
Author(s):  
David Marker
Keyword(s):  
Finite Union ◽  
Structure Theory ◽  
Real Closed Fields ◽  
Minimal Theory ◽  
Order Language ◽  
Closed Fields ◽  
Hanf Number ◽  
Definable Sets ◽  
Omitting Types ◽  
Binary Relation Symbol

Let L be a first order language containing a binary relation symbol <.Definition. Suppose ℳ is an L-structure and < is a total ordering of the domain of ℳ. ℳ is ordered minimal (-minimal) if and only if any parametrically definable X ⊆ ℳ can be represented as a finite union of points and intervals with endpoints in ℳ.In any ordered structure every finite union of points and intervals is definable. Thus the -minimal structures are the ones with no unnecessary definable sets. If T is a complete L-theory we say that T is strongly (-minimal if and only if every model of T is -minimal.The theory of real closed fields is the canonical example of a strongly -minimal theory. Strongly -minimal theories were introduced (in a less general guise which we discuss in §6) by van den Dries in [1]. Extending van den Dries' work, Pillay and Steinhorn (see [3], [4] and [2]) developed an extensive structure theory for definable sets in strongly -minimal theories, generalizing the results for real closed fields. They also established several striking analogies between strongly -minimal theories and ω-stable theories (most notably the existence and uniqueness of prime models). In this paper we will examine the construction of models of strongly -minimal theories emphasizing the problems involved in realizing and omitting types. Among other things we will prove that the Hanf number for omitting types for a strongly -minimal theory T is at most (2∣T∣)+, and characterize the strongly -minimal theories with models order isomorphic to (R, <).

scholarly journals The nonlocal solvability conditions for a system of two quasilinear equations of the first order with absolute terms

2019 ◽  
Vol 21 (3) ◽  
pp. 317-328
Author(s):  
Marina V. Dontsova
Keyword(s):  
Cauchy Problem ◽  
Finite Number ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Local Solution ◽  
Quasilinear Equations ◽  
Solvability Conditions ◽  
First Order ◽  
The Cauchy Problem ◽  
Global Estimates

The Cauchy problem for a system of two first-order quasilinear equations with absolute terms is considered. The study of this problem’s solvability in original coordinates is based on the method of an additional argument. The existence of the local solution of the problem with smoothness which is not lower than the smoothness of the initial conditions, is proved. Sufficient conditions of existence are determined for the nonlocal solution that is continued by a finite number of steps from the local solution. The proof of the nonlocal resolvability of the Cauchy problem relies on original global estimates.

On first-order sentences without finite models

2004 ◽  
Vol 69 (2) ◽  
pp. 329-339 ◽  
Author(s):  
Marko Djordjević
Keyword(s):  
Binary Relation ◽  
Function Symbol ◽  
Complete Theory ◽  
Order Property ◽  
Unary Function ◽  
First Order ◽  
Finite Models

We will mainly be concerned with a result which refutes a stronger variant of a conjecture of Macpherson about finitely axiomatizable ω-categorical theories. Then we prove a result which implies that the ω-categorical stable pseudoplanes of Hrushovski do not have the finite submodel property.Let's call a consistent first-order sentence without finite models an axiom of infinity. Can we somehow describe the axioms of infinity? Two standard examples are:ϕ1: A first-order sentence which expresses that a binary relation < on a nonempty universe is transitive and irreflexive and that for every x there is y such that x < y.ϕ2: A first-order sentence which expresses that there is a unique x such that, (0) for every y, s(y) ≠ x (where s is a unary function symbol),and, for every x, if x does not satisfy (0) then there is a unique y such that s(y) = x.Every complete theory T such that ϕ1 ϵ T has the strict order property (as defined in [10]), since the formula x < y will have the strict order property for T. Let's say that if Ψ is an axiom of infinity and every complete theory T with Ψ ϵ T has the strict order property, then Ψ has the strict order property.Every complete theory T such that ϕ2 ϵ T is not ω-categorical. This is the case because a complete theory T without finite models is ω-categorical if and only if, for every 0 < n < ω, there are only finitely many formulas in the variables x1,…,xn, up to equivalence, in any model of T.

scholarly journals Spatio-temporal patterns with hyperchaotic dynamics in diffusively coupled biochemical oscillators

1997 ◽  
Vol 1 (2) ◽  
pp. 161-167 ◽  
Author(s):  
Gerold Baier ◽  
Sven Sahle
Keyword(s):  
Dynamical Systems ◽  
Finite Number ◽  
Temporal Patterns ◽  
Autocatalytic Reaction ◽  
Living Systems ◽  
First Order ◽  
Autocatalytic Reactions ◽  
Complex Patterns ◽  
Spatio Temporal

We present three examples how complex spatio-temporal patterns can be linked to hyperchaotic attractors in dynamical systems consisting of nonlinear biochemical oscillators coupled linearly with diffusion terms. The systems involved are: (a) a two-variable oscillator with two consecutive autocatalytic reactions derived from the Lotka–Volterra scheme; (b) a minimal two-variable oscillator with one first-order autocatalytic reaction; (c) a three-variable oscillator with first-order feedback lacking autocatalysis. The dynamics of a finite number of coupled biochemical oscillators may account for complex patterns in compartmentalized living systems like cells or tissue, and may be tested experimentally in coupled microreactors.

Asymptotic behavior of Hill's estimate and applications

1986 ◽  
Vol 23 (04) ◽  
pp. 922-936
Author(s):  
Gane Samb Lo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Finite Number ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Complete Theory ◽  
General Distribution ◽  
Stable Law ◽  
Biological Phenomena ◽  
Pareto’S Law

The problem of estimating the exponent of a stable law is receiving an increasing amount of attention because Pareto's law (or Zipf's law) describes many biological phenomena very well (see e.g. Hill (1974)). This problem was first solved by Hill (1975), who proposed an estimate, and the convergence of that estimate to some positive and finite number was shown to be a characteristic of distribution functions belonging to the Fréchet domain of attraction (Mason (1982)). As a contribution to a complete theory of inference for the upper tail of a general distribution function, we give the asymptotic behavior (weak and strong) of Hill's estimate when the associated distribution function belongs to the Gumbel domain of attraction. Examples, applications and simulations are given.

Theories with finitely many models

1986 ◽  
Vol 51 (2) ◽  
pp. 374-376 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Natural Number ◽  
Complete Theory ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Simple Observation ◽  
Finite Structure ◽  
Order Language ◽  
Complete Theories

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x1,…,xn. While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln. This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.Theorem. For each n ≤ 3 there are complete theories in L2n−2andL2n−1having exactly n + 1 models.In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln, we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a1, b1, …, an, bn. On each move, player I picks up any of the counters and places it on an element of the appropriate structure.

On the second spectrum and the second classical Zariski topology of a module

2015 ◽  
Vol 14 (10) ◽  
pp. 1550150 ◽  
Author(s):  
Seçil Çeken ◽  
Mustafa Alkan
Keyword(s):  
Finite Number ◽  
Associative Ring ◽  
Finite Union ◽  
Zariski Topology ◽  
Topological Properties ◽  
Noetherian Module ◽  
Algebraic Properties ◽  
Second Spectrum

Let R be an associative ring with identity and Specs(M) denote the set of all second submodules of a right R-module M. In this paper, we investigate some interrelations between algebraic properties of a module M and topological properties of the second classical Zariski topology on Specs(M). We prove that a right R-module M has only a finite number of maximal second submodules if and only if Specs(M) is a finite union of irreducible closed subsets. We obtain some interrelations between compactness of the second classical Zariski topology of a module M and finiteness of the set of minimal submodules of M. We give a connection between connectedness of Specs(M) and decomposition of M for a right R-module M. We give several characterizations of a noetherian module M over a ring R such that every right primitive factor of R is artinian for which Specs(M) is connected.

scholarly journals Generalized prime models

1971 ◽  
Vol 36 (4) ◽  
pp. 593-606 ◽  
Author(s):  
Robert Fittler
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
First Order ◽  
First Order Theory ◽  
General Conditions

A prime model O of some complete theory T is a model which can be elementarily imbedded into any model of T (cf. Vaught [7, Introduction]). We are going to replace the assumption that T is complete and that the maps between the models of T are elementary imbeddings (elementary extensions) by more general conditions. T will always be a first order theory with identity and may have function symbols. The language L(T) of T will be denumerable. The maps between models will be so called F-maps, i.e. maps which preserve a certain set F of formulas of L(T) (cf. I.1, 2). Roughly speaking a generalized prime model of T is a denumerable model O which permits an F-map O→M into any model M of T. Furthermore O has to be “generated” by formulas which belong to a certain subset G of F.

Asymptotic behavior of Hill's estimate and applications

1986 ◽  
Vol 23 (4) ◽  
pp. 922-936 ◽  
Author(s):  
Gane Samb Lo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Finite Number ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Complete Theory ◽  
General Distribution ◽  
Stable Law ◽  
Biological Phenomena ◽  
Pareto’S Law

The problem of estimating the exponent of a stable law is receiving an increasing amount of attention because Pareto's law (or Zipf's law) describes many biological phenomena very well (see e.g. Hill (1974)). This problem was first solved by Hill (1975), who proposed an estimate, and the convergence of that estimate to some positive and finite number was shown to be a characteristic of distribution functions belonging to the Fréchet domain of attraction (Mason (1982)). As a contribution to a complete theory of inference for the upper tail of a general distribution function, we give the asymptotic behavior (weak and strong) of Hill's estimate when the associated distribution function belongs to the Gumbel domain of attraction. Examples, applications and simulations are given.

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