scholarly journals A note on Lascar strong types in simple theories

1998 ◽  
Vol 63 (3) ◽  
pp. 926-936 ◽  
Author(s):  
Byunghan Kim
Keyword(s):  
Simple Theory ◽  
Strong Type ◽  
Simple Theories

AbstractLet T be a countable, small simple theory. In this paper, we prove that for such T, the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.

Definability in low simple theories

2000 ◽  
Vol 65 (4) ◽  
pp. 1481-1490 ◽  
Author(s):  
Ziv Shami
Keyword(s):  
Amalgamation Property ◽  
Simple Theory ◽  
Weak Version ◽  
Strong Type ◽  
Simple Theories ◽  
New Class ◽  
Bounded Set ◽  
Almost All ◽  
Independence Theorem ◽  
Mutually Independent

In 1978 Shelah introduced a new class of theories, called simple (see [Shi]) which properly contained the class of stable theories. Shelah generalized part of the theory of forking to the simple context. After approximately 15 years of neglecting the general theory (although there were works by Hrushovski on finite rank with a definability assumption, as well as deep results on specific simple theories by Cherlin, Hrushovski, Chazidakis, Macintyre and Van den dries, see [CH], [CMV], [HP1], [HP2], [ChH]) there was a breakthrough, initiated with the work of Kim ([K1]). Kim proved that almost all the technical machinery of forking developed in the stable context could be generalized to simple case. However, the theory of multiplicity (i.e., the description of the (bounded) set of non forking extensions of a given complete type) no longer holds in the context of simple theories. Indeed, by contrast to simple theories, stable theories share a strong amalgamation property of types, namely if p and q are two “free” complete extensions over a superset of A, and there is no finite equivalence relation over A which separates them, then the conjunction of p and q is consistent (and even free over A.) In [KP] Kim and Pillay proved a weak version of this property for any simple theory, namely “the Independence Theorem for Lascar strong types”. This was a weaker version both because of the requirement that the sets of parameters of the types be mutually independent, as well as the use of Lascar strong types instead of the usual strong types. A very fundamental and interesting problem is whether the independence theorem can be proved for any simple theory, using only the usual strong types. In 1997 Buechler proved ([Bu]) the strong-type version of the independence theorem for an important subclass of simple theories, namely the class of low theories (which includes the class of stable theories and the class of supersimple theories of finite D-rank.)

scholarly journals On almost orthogonality in simple theories

2004 ◽  
Vol 69 (2) ◽  
pp. 398-408 ◽  
Author(s):  
Itay Ben-Yaacov ◽  
Frank O. Wagner
Keyword(s):  
Fundamental System ◽  
Simple Theory ◽  
Strong Type ◽  
Finitely Generated ◽  
Simple Theories ◽  
Real Type

Abstract.1. We show that if p is a real type which is internal in a set Σ of partial types in a simple theory, then there is a type p′ interbounded with p, which is finitely generated over Σ, and possesses a fundamental system of solutions relative to Σ.2. If p is a possibly hyperimaginary Lascar strong type, almost Σ-internal, but almost orthogonal to Σω, then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts on p while fixing Σ generically In case p is Σ-internal and T is stable, this is the binding group of p over Σ.

On Kueker simple theories

2005 ◽  
Vol 70 (1) ◽  
pp. 216-222 ◽  
Author(s):  
Ziv Shami
Keyword(s):  
Simple Theory ◽  
Simple Theories

AbstractWe show that a Kueker simple theory eliminates ∃∞ and densely interprets weakly minimal formulas. As part of the proof we generalize Hrushovski's dichotomy for almost complete formulas to simple theories. We conclude that in a unidimensional simple theory an almost-complete formula is either weakly minimal or trivially-almost-complete. We also observe that a small unidimensional simple theory is supersimple of finite SU-rank.

Properties and consequences of Thorn-independence

2006 ◽  
Vol 71 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Alf Onshuus
Keyword(s):  
Large Class ◽  
Geometric Structure ◽  
Simple Theory ◽  
Simple Theories

AbstractWe develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure.We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.

Coordinatisation by binding groups and unidimensionality in simple theories

2004 ◽  
Vol 69 (4) ◽  
pp. 1221-1242 ◽  
Author(s):  
Ziv Shami
Keyword(s):  
Finite Sequence ◽  
Simple Theory ◽  
Simple Theories ◽  
Infinite Type ◽  
Definable Set

Abstract.In a simple theory with elimination of finitary hyperimaginaries if tp(a) is real and analysable over a definable set Q, then there exists a finite sequence (ai \ i ≤ n*) ⊆ dcleq(a) with an* = a such that for every i ≤ n* if pi = tp(ai/{aj |j < i}) then Aut(pi / Q) is type-definable with its action on . A unidimensional simple theory eliminates the quantifier ∃∞ and either interprets (in Ceq) an infinite type-definable group or has the property that ACL(Q) = C for every infinite definable set Q.

On types of CB-rank 1 in simple theories

2008 ◽  
Vol 7 (4) ◽  
pp. 895-899 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Recent Result ◽  
Simple Theory ◽  
Countable Model ◽  
Stable Case ◽  
Simple Theories ◽  
Complete Type ◽  
Special Case

AbstractWe prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor–Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T has infinitely many countable elementary extensions up to M0-isomorphism. The latter extends earlier results of the author in the stable case, and is a special case of a recent result of Tanovic.

Lascar strong types in some simple theories

1999 ◽  
Vol 64 (2) ◽  
pp. 817-824 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Strong Type ◽  
Simple Theories ◽  
Low Theory

AbstractIn this paper a class of simple theories, called the low theories is developed, and the following is proved.Theorem. Let T be a low theory, A a set and a, b elements realizing the same strong type over A. Then, a and b realize the same Lasear strong type over A.

scholarly journals AN INDEPENDENCE THEOREM FOR NTP2 THEORIES

2014 ◽  
Vol 79 (01) ◽  
pp. 135-153 ◽  
Author(s):  
ITAÏ BEN YAACOV ◽  
ARTEM CHERNIKOV
Keyword(s):  
Chain Condition ◽  
Strong Type ◽  
Simple Theories ◽  
Independence Theorem

Abstract We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP2 theory. We also define the dividing order of a theory—a generalization of Poizat’s fundamental order from stable theories—and give some equivalent characterizations under the assumption of NTP2. The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.

HYPERDEFINABLE GROUPS IN SIMPLE THEORIES

2001 ◽  
Vol 01 (01) ◽  
pp. 125-172 ◽  
Author(s):  
FRANK WAGNER
Keyword(s):  
Simple Theory ◽  
Basic Theory ◽  
Simple Theories ◽  
Group Quotient ◽  
General Kind

We study hyperdefinable groups, the most general kind of groups interpretable in a simple theory. After developing their basic theory, we prove the appropriate versions of Hrushovski's group quotient theorem and the Weil–Hrushovski group chunk theorem. We also study locally modular hyperdefinable groups and prove that they are bounded-by-Abelian-by-bounded. Finally, we analyze hyperdefinable groups in supersimple theories.

scholarly journals A CLASSIFICATION OF 2-CHAINS HAVING 1-SHELL BOUNDARIES IN ROSY THEORIES

2015 ◽  
Vol 80 (1) ◽  
pp. 322-340 ◽  
Author(s):  
BYUNGHAN KIM ◽  
SUNYOUNG KIM ◽  
JUNGUK LEE
Keyword(s):  
Upper Bound ◽  
Homology Group ◽  
Strong Type ◽  
Simple Theories ◽  
Shell Boundary ◽  
Rosy Theories

AbstractWe classify, in a nontrivial amenable collection of functors, all 2-chains up to the relation of having the same 1-shell boundary. In particular, we prove that in a rosy theory, every 1-shell of a Lascar strong type is the boundary of some 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is no upper bound on the minimal lengths of 2-chains whose boundary is a 1-shell.

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