Results 1  10
of
11
SIMPLE HOMOGENEOUS MODELS
, 2002
"... Geometrical stability theory is a powerful set of modeltheoretic tools that can lead to structural results on models of a simple firstorder theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theor ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
Geometrical stability theory is a powerful set of modeltheoretic tools that can lead to structural results on models of a simple firstorder theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theory (a saturated model) in which the Stone space topology on ultrafilters of definable relations is compact. Here we operate in the more general setting of homogeneous models, which typically have noncompact Stone topologies. A structure M equipped with a class of finitary relations R is strongly λ−homogeneous if orbits under automorphisms of (M, R) have finite character in the following sense: Given α an ordinal < λ ≤ M  and sequences ā = { ai: i < α}, ¯ b = { bi: i < α} from M, if (ai1,..., ain) and (bi1,..., bin) have the same orbit, for all n and i1 < · · · < in < α, then f(ā) = ¯ b for some automorphism f of (M, R). In this paper strongly λ−homogeneous models (M, R) in which the elements of R induce a symmetric and transitive notion of independence with bounded character are studied. This notion of independence, defined using a combinatorial condition called “dividing”, agrees with forking independence when (M, R) is saturated. The concept central to the development of geometrical stability theory for saturated structures, namely the canonical base, is also shown to exist in this setting. These results broaden the scope of the
Lascar Strong Types in Some Simple Theories
, 1997
"... In 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 Lascar strong type over A . The reader is expected to be famil ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
In 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 Lascar strong type over A . The reader is expected to be familiar with forking in simple theories, as developed in Kim's thesis [Kim]. The Lascar strong type of a over A is denoted lstp(a=A) . Unless stated otherwise, we work in the context of a simple theory in this paper. 1 Amalgamation properties Type amalgamation (the Independence Theorem) is perhaps the most useful property of forking dependence in a simple theory. First, we stress an important fact from [Kim]. Lemma 1.1 Let A be a set, a; b elements such that lstp(a=A) = lstp(b=A) and a j
SUPERSIMPLE FIELDS AND DIVISION RINGS
 MATHEMATICAL RESEARCH LETTERS
, 1998
"... It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple. ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
From Stability To Simplicity
 Bulletin of Symbolic Logic 4
, 1998
"... this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others. ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.
Simplicity, And Stability In There
 JOURNAL OF SYMBOLIC LOGIC
, 1999
"... Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T , canonical base of ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T , canonical base of an amalgamation class P is the union of names of /definitions of P , / ranging over stationary Lformulas in P . Also, we prove that the same is true with stable formulas for an 1based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.
Some remarks on indiscernible sequences
 Mathematical Logic Quarterly
"... We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of coheirs.
Adding Skolem functions to simple theories
 Arch. Math. Logic
"... We examine the conditions under which we can keep simplicity or categoricity after adding a Skolem function to the theory. AMS classification: 03C45, 03C50 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We examine the conditions under which we can keep simplicity or categoricity after adding a Skolem function to the theory. AMS classification: 03C45, 03C50
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE ITAY BENYAACOV
, 902
"... Abstract. We prove that for every simple theory T (or even simple thick compact abstract theory) there is a (unique) compact abstract theory T P whose saturated models are the lovely pairs of T. Independencetheoretic results that were proved in [BPV03] when T P is a first order theory are proved fo ..."
Abstract
 Add to MetaCart
Abstract. We prove that for every simple theory T (or even simple thick compact abstract theory) there is a (unique) compact abstract theory T P whose saturated models are the lovely pairs of T. Independencetheoretic results that were proved in [BPV03] when T P is a first order theory are proved for the general case: in particular T P is simple and we characterise independence.