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The Lascar Group and the Strong Types of Hyperimaginaries

Byunghan Kim
2013 Notre Dame Journal of Formal Logic  
This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries, and make some new observations on the strong types over those.  ...  In particular we show that in a simple theory, Ltp ≡ stp in real context implies that for hyperimaginary context.  ...  Notably in [9] [1], a new look on the Lascar group is given, and using compact Lie group theory Lascar and Pillay proved that any bounded hyperimaginary is interdefinable with a sequence of finitary bounded  ... 
doi:10.1215/00294527-2143916 fatcat:yo6veora5befhjeppvdjpfvd5y

Automorphism groups over a hyperimaginary [article]

Byunghan Kim, Hyoyoon Lee
2021 arXiv   pre-print
On the other, we extend Newelski's Theorem in "The diameter of a Lascar strong type" [12] that 'a G-compact theory over a set has a uniform bound for the Lascar distances' to the hyperimaginary context  ...  We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable.  ...  Namely we investigate the Kim-Pillay group and the Shelah group, and corresponding notions of Kim-Pillay types and Shelah strong types, over a hyperimaginary.  ... 
arXiv:2106.09200v1 fatcat:b2l3twzqzngwnie2hz57befi2u

Elimination of Hyperimaginaries and Stable Independence in Simple CM-Trivial Theories

D. Palacín, F. O. Wagner
2013 Notre Dame Journal of Formal Logic  
In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries.  ...  Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.  ...  Since h ∈ bdd(c), equality of Lascar strong types over c refines equality of Lascar strong types over h, and the class of a modulo the former is clearly in dcl(ac).  ... 
doi:10.1215/00294527-2143970 fatcat:tf2snizezjgxliqdu36xejjb4a

Normal hyperimaginaries [article]

Enrique Casanovas, Joris Potier
2013 arXiv   pre-print
We introduce the notion of normal hyperimaginary and we develop its basic theory.  ...  In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of φ-types for bounded hyperimaginaries.  ...  It is also called the Kim-Pillay equivalence relation and its classes are called KP-strong types. The set of all KP-classes of α-tuples is C α /KP.  ... 
arXiv:1112.2049v3 fatcat:uryyxpa2ljfhhfyjzgwtvpk2ua

The Group Configuration in Simple Theories and its Applications

Itay Ben-Yaacov, Ivan Tomašić, Frank O. Wagner
2002 Bulletin of Symbolic Logic  
of pseudolinearity and local modularity.The proof necessitated an extension of the model-theoretic framework to includealmost hyperimaginaries, and the study ofpolygroups.  ...  theorem, or, in the ω-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence  ...  Two ultraimaginaries a R and b R have the same (Lascar strong) type over a hyperimaginary c if there are representatives which do.  ... 
doi:10.2178/bsl/1182353874 fatcat:7ed2ljvawrez3irfi2wza63uxu

Group configurations and germs in simple theories

Itay Ben-Yaacov
2002 Journal of Symbolic Logic (JSL)  
This is the first step towards the proof of the group configuration theorem for simple theories, which is completed in [3].  ...  We develop the theory of germs of generic functions in simple theories.  ...  set of all possible extensions of the graph to a Lascar strong type over the parameter.  ... 
doi:10.2178/jsl/1190150301 fatcat:i5xr27zoqvff7i6qmv4m5kzwte

Page 757 of Mathematical Reviews Vol. , Issue 99b [page]

1991 Mathematical Reviews  
The conclusion is a set of four open questions, the last one asking whether, in a simple theory, the Lascar strong type is nothing more than the strong type, and a hyperimaginary nothing more than a (possibly  ...  Another corollary is the same property over an arbitrary set of parameters inside a Lascar strong type (if a and b have the same type over a model containing A, then they have the same Lascar strong type  ... 

Supersimple Theories [chapter]

Frank O. Wagner
2000 Simple Theories  
In fact the reader should note that our proof also yields the equality of Lascar strong type with strong type in low simple theories.  ...  consider type-definable equivalence relations with a small number of classes (corresponding to Lascar strong types).  ... 
doi:10.1007/978-94-017-3002-0_5 fatcat:clqqg5w64vfqhctaq2aj7ltb4a

Generalized amalgamation and homogeneity [article]

Daniel Palacín
2016 arXiv   pre-print
In particular, this adapts a result of Koponen for binary homogeneous structures to arbitrary ones without binary relations.  ...  Furthermore, we point out a relation between generalized amalgamation, triviality and quantifier elimination in simple theories.  ...  This motivates the definition of a Lascar strong type: A type over which the Independence Theorem holds. We denote the Lascar strong type of a over A by Lstp(a/A).  ... 
arXiv:1603.09694v3 fatcat:gqrza3hkeneonix5qbd77wymde

Supersimple theories

Steven Buechler, Anand Pillay, Frank Wagner
2000 Journal of The American Mathematical Society  
In fact the reader should note that our proof also yields the equality of Lascar strong type with strong type in low simple theories.  ...  consider type-definable equivalence relations with a small number of classes (corresponding to Lascar strong types).  ... 
doi:10.1090/s0894-0347-00-00350-7 fatcat:c7drkevsbjcylpghdoi3wrtu6m

APPLICATIONS OF THE GROUP CONFIGURATION THEOREM IN SIMPLE THEORIES

IVAN TOMAŠIĆ, FRANK O. WAGNER
2003 Journal of Mathematical Logic  
We reconstruct the group action in the group configuration theorem.  ...  We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective  ...  This coincides with the definitions for hyperimaginaries; two almost hyperimaginaries have the same (Lascar strong) type over c if and only if they are conjugate under a (strong) c-automorphism.  ... 
doi:10.1142/s0219061303000261 fatcat:kkgqzew6andhvknpeqnzqk5xlq

Plus ultra [article]

Frank Olaf Wagner
2014 arXiv   pre-print
Finally, we prove some uniform versions of the weak canonical base property.  ...  We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts.  ...  of Lascar strong types over A of real tuples of length κ, since equality of Lascar strong type over A is the finest bounded A-invariant equivalence relation.  ... 
arXiv:1201.1114v4 fatcat:e6nek4gii5hbzkwlfy2naar5mu

Bounded ultraimaginary independence and its total Morley sequences [article]

James Hanson
2022 arXiv   pre-print
sequence over A if and only if whenever I and I' have the same Lascar strong type over A, I and I' are related by the transitive, symmetric closure of the relation 'J+K is A-indiscernible.'  ...  set of hyperimaginaries A and ultraimaginaries b and c, there is a b' ≡_A b such that b' _Amu c).  ...  Another appealing aspect of ultraimaginaries is that they characterize Lascar strong type in the same way that hyperimaginaries characterize Kim-Pillay strong type.  ... 
arXiv:2201.03631v1 fatcat:qdlpt3bnozbjlha2rvr52pvmai

On almost orthogonality in simple theories

Itay Ben-Yaacov, Frank O. Wagner
2004 Journal of Symbolic Logic (JSL)  
fixing Σ generically In case p is Σ-internal and T is stable, this is the binding group of p over Σ.  ...  If p is a possibly hyperimaginary Lascar strong type, almost Σ-internal, but almost orthogonal to Σ ω , then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts on p while  ...  to a Lascar strong type).  ... 
doi:10.2178/jsl/1082418533 fatcat:m55czfzw6jdvpjlse6zenhewgq

Rigidity, internality and analysability

Daniel Palacín, Frank O. Wagner
2014 Fundamenta Mathematicae  
We prove a version of Hrushovski's Socle Lemma for rigid groups in an arbitrary simple theory. 2000 Mathematics Subject Classification. 03C45.  ...  Let G be a rigid abelian group of finite SU-rank, and p a Lascar strong G-type of finite stabilizer. Then p is almost internal in the set Σ of all types of SU-rank one.  ...  More generally, Chatzidakis proves the following two facts (she works in a simple theory with elimination of hyperimaginaries and the family Σ of all non locally modular (Lascar) strong types of SU-rank  ... 
doi:10.4064/fm224-1-3 fatcat:vppumaiuvjcmpotj664327yr5e
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