Flat Morley sequences

1999 ◽  
Vol 64 (3) ◽  
pp. 1261-1279
Author(s):  
Ludomir Newelski
Keyword(s):  
Superstable Theory ◽  
Complete Type ◽  
Finite Set ◽  
Countable Models

AbstractAssume T is a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in a type p, in case when p is a complete type over a finite set of parameters. We show that for any flat Morley sequence Q there is a model M of T which is τ-atomic over {Q}. When additionally T has few countable models and is 1-based, we prove that within M there is an infinite Morley sequence I, with I ⊂ dcl(Q), such that M is prime over I.

A note on subgroups of the automorphism group of a saturated model, and regular types

1989 ◽  
Vol 54 (3) ◽  
pp. 858-864 ◽  
Author(s):  
A. Pillay
Keyword(s):  
Automorphism Group ◽  
Algebraic Closure ◽  
Saturated Model ◽  
Superstable Theory ◽  
Alternative Characterization ◽  
Finite Set ◽  
Superstable Theories

AbstractLet M be a saturated model of a superstable theory and let G = Aut(M). We study subgroups H of G which contain G(A), A the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types p in the context of p-simple types.

Isolated types in a weakly minimal set

1987 ◽  
Vol 52 (2) ◽  
pp. 543-547 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Prime Model ◽  
Morley Rank ◽  
Minimal Set ◽  
Superstable Theory ◽  
Finite Set ◽  
Minimal Type ◽  
Minimal Formula

AbstractTheorem A. Let T be a small superstable theory, A a finite set, and ψ a weakly minimal formula over A which is contained in some nontrivial type which does not have Morley rank. Then ψ is contained in some nonalqebraic isolated type over A.As an application we proveTheorem B. Suppose that T is small and superstable, A is finite, and there is a nontrivial weakly minimal type p ∈ S(A) which does not have Morley rank. Then the prime model over A is not minimal over A.

Geometry of *-Finite Types

1999 ◽  
Vol 64 (4) ◽  
pp. 1375-1395 ◽  
Author(s):  
Ludomir Newelski
Keyword(s):  
Algebraic Group ◽  
Division Ring ◽  
Algebraic Groups ◽  
Superstable Theory ◽  
Countable Models

AbstractAssume T is a superstable theory with < 2ℵ0 countable models. We prove that any *- algebraic type of -rank > 0 is m-nonorthogonal to a *-algebraic type of -rank 1. We study the geometry induced by m-dependence on a *-algebraic type p* of -rank 1. We prove that after some localization this geometry becomes projective over a division ring . Associated with p* is a meager type p. We prove that p is determined by p* up to nonorthogonality and that underlies also the geometry induced by forking dependence on any stationarization of p. Also we study some *-algebraic *-groups of -rank 1 and prove that any *-algebraic *-group of -rank 1 is abelian-by-finite.

Omitting models

1977 ◽  
Vol 42 (1) ◽  
pp. 29-32
Author(s):  
Ernest Snapper
Keyword(s):  
Homogeneous Model ◽  
Countable Model ◽  
Complete Theory ◽  
Finite Set ◽  
Homogeneous Models ◽  
Definition Of ◽  
General Aspects ◽  
The Universe ◽  
Main Definition ◽  
Countable Models

The purpose of this paper is to introduce the notion of “omitting models” and to derive a very natural theorem concerning it (Theorem 1). A corollary of this theorem is the remarkable theorem of Vaught [3] which states that a countable complete theory cannot have precisely two nonisomorphic countable models. In fact, we show that our theorem implies Rosenstein's theorem [2] which, in turn, implies Vaught's theorem.T stands for a countable complete theory whose (countable) language is denoted by L. Following [1], a countably homogeneous model of T is a countable model of T with the property that, for any two n-tuples a1, …, an and b1,…,bn of the universe of whose types are the same, there is an automorphism of which maps ai, on bi, for i = 1, …, n [1, p. 129 and Proposition 3.2.9, p. 131]. “Homogeneous model” always means “countably homogeneous model.” “Type of T” always stands for “n-type of T” where n ≥ s 0, i.e., for the type of some n-tuple of individuals of the universe of some model of T. We often use that two homogeneous models which realize the same types are isomorphic [1, Proposition 3.2.9, p. 131].It is well known that every type of T is realized by at least one countable model of T. The main definition of this paper is:Definition 1. A set of countable models of T is omissible or “may be omitted” if every type of T is realized by at least one countable model of T which is not isomorphic to a model in the set.The main theorem of the paper is:Theorem 1. If a countable complete theory is not ω-categorical, every finite set of its homogeneous models may be omitted.The theorem is proved in §1 and in §2 it is shown how Vaught's and Rosenstein's theorems follow from it. §3 discusses some general aspects of omitting models.

Limit ultrapowers and abstract logics

1987 ◽  
Vol 52 (2) ◽  
pp. 437-454 ◽  
Author(s):  
Paolo Lipparini
Keyword(s):  
Interpolation Property ◽  
Finite Cover ◽  
Abstract Logic ◽  
Superstable Theory ◽  
Proper Extension ◽  
Elementary Classes ◽  
Abstract Logics ◽  
Countable Models

AbstractWe associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L.For every countably generated [ω, ω]-compact logic L, our main applications are:(i) Elementary classes of L can be characterized in terms of ≡L only.(ii) If and are countable models of a countable superstable theory without the finite cover property, then .(iii) There exists the “largest” logic M such that complete extensions in the sense of M and L are the same; moreover M is still [ω, ω]-compact and satisfies an interpolation property stronger than unrelativized ⊿-closure.(iv) If L = Lωω(Qx), then cf(ωx) > ω and λω < ωx, for all λ < ωx.We also prove that no proper extension of Lωω generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning Lκλ-compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics.

The classification of small weakly minimal sets. II

1988 ◽  
Vol 53 (2) ◽  
pp. 625-635 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Stable Theory ◽  
Minimal Set ◽  
Minimal Sets ◽  
Finite Set ◽  
Unidimensional Theory ◽  
The Universe ◽  
Countable Models

AbstractThe main result is Vaught's conjecture for weakly minimal, locally modular and non-ω-stable theories. The more general results yielding this are the following.Theorem A. Suppose that T is a small unidimensional theory and D is a weakly minimal set, definable over the finite set B. Then for all finite A ⊂ D there are only finitely many nonalgebraic strong types over B realized in acl(A) ∩ D.Theorem B. Suppose that T is a small, unidimensional, non-ω-stable theory such that the universe is weakly minimal and locally modular. Then for all finite A there is a finite B ⊂ cl(A) such that a ∈ cl(A) iff a ∈ cl(b) for some b ∈ B.Recall the property (S) defined in the abstract of [B1].Theorem C. Let T be as in Theorem B. Then, if T does not satisfy (S), T hasmany countable models.Combining Theorem C and the results in [B1] we obtain Vaught's conjecture for such theories.

Non-repetitive sequences

1970 ◽  
Vol 68 (2) ◽  
pp. 267-274 ◽  
Author(s):  
P. A. B. Pleasants
Keyword(s):  
Repetitive Sequences ◽  
Finite Set ◽  
Infinite Sequences

This note is concerned with infinite sequences whose terms are chosen from a finite set of symbols. A segment of such a sequence is a set of one or more consecutive terms, and a repetition is a pair of finite segments that are adjacent and identical. A non-repetitive sequence is one that contains no repetitions.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

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