Definability in low simple theories

2000 ◽  
Vol 65 (4) ◽  
pp. 1481-1490 ◽  
Author(s):  
Ziv Shami

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.)

2014 ◽  
Vol 79 (01) ◽  
pp. 135-153 ◽  
Author(s):  
ITAÏ BEN YAACOV ◽  
ARTEM CHERNIKOV

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.


2004 ◽  
Vol 69 (2) ◽  
pp. 398-408 ◽  
Author(s):  
Itay Ben-Yaacov ◽  
Frank O. Wagner

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.


1998 ◽  
Vol 63 (3) ◽  
pp. 926-936 ◽  
Author(s):  
Byunghan Kim

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.


1994 ◽  
Vol 3 (4) ◽  
pp. 435-454 ◽  
Author(s):  
Neal Brand ◽  
Steve Jackson

In [11] it is shown that the theory of almost all graphs is first-order complete. Furthermore, in [3] a collection of first-order axioms are given from which any first-order property or its negation can be deduced. Here we show that almost all Steinhaus graphs satisfy the axioms of almost all graphs and conclude that a first-order property is true for almost all graphs if and only if it is true for almost all Steinhaus graphs. We also show that certain classes of subgraphs of vertex transitive graphs are first-order complete. Finally, we give a new class of higher-order axioms from which it follows that large subgraphs of specified type exist in almost all graphs.


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

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.


2014 ◽  
Vol 53 (2) ◽  
pp. R47-R59 ◽  
Author(s):  
Subhamoy Dasgupta ◽  
Bert W O'Malley

Transcriptional coactivators have evolved as an important new class of functional proteins that participate with virtually all transcription factors and nuclear receptors (NRs) to intricately regulate gene expression in response to a wide variety of environmental cues. Recent findings have highlighted that coactivators are important for almost all biological functions, and consequently, genetic defects can lead to severe pathologies. Drug discovery efforts targeting coactivators may prove valuable for treatment of a variety of diseases.


2006 ◽  
Vol 71 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Alf Onshuus

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.


2004 ◽  
Vol 69 (4) ◽  
pp. 1221-1242 ◽  
Author(s):  
Ziv Shami

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.


2020 ◽  
Author(s):  
Abdullahi Aborode ◽  
Kubeyinje Winner ◽  
Oni Ebenezer Ayomide

A new class of corona virus, known as SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2) has been found to be responsible for occurrence of this disease. As far as the history of human civilization is concerned there are instances of severe outbreaks of diseases caused by a number of viruses. According to the report of the World Health Organization (WHO as of June 5, 2020), the current pandemic of COVID-19 has affected 6,749,371 people, 3,277, 596 recovered and killed 394,527 people in 215 countries throughout the world. Till now there is no report of any clinically approved antiviral drugs or vaccines that are effective against COVID-19. It has rapidly spread around the world, posing enormous health, economic, environmental and social challenges to the entire human population. The coronavirus outbreak is severely disrupting the global economy. Almost all the nations are struggling to slow down the transmission of the disease by testing and treating patients, quarantining suspected persons through contact tracing, restricting large gatherings, maintaining complete or partial lock down etc. This paper describes the effects of COVID-19 on society and global environment, and the possible ways in which the disease can be prevented or controlled.


2008 ◽  
Vol 7 (4) ◽  
pp. 895-899 ◽  
Author(s):  
Anand Pillay

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.


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