scholarly journals Reconstructing structures with the strong small index property up to bi-definability

2019 ◽  
Vol 247 (1) ◽  
pp. 25-35
Author(s):  
Gianluca Paolini ◽  
Saharon Shelah
Keyword(s):  
Small Index Property ◽  
Small Index

scholarly journals The Small Index Property for ω-Stable (ω-Categorical Structures and for the Random Graph

1993 ◽  
Vol s2-48 (2) ◽  
pp. 204-218 ◽  
Author(s):  
Wilfrid Hodges ◽  
Ian Hodkinson ◽  
Daniel Lascar ◽  
Saharon Shelah
Keyword(s):  
Random Graph ◽  
Small Index Property ◽  
Small Index

scholarly journals The small index property for homogeneous models in AEC’s

2017 ◽  
Vol 57 (1-2) ◽  
pp. 141-157
Author(s):  
Zaniar Ghadernezhad ◽  
Andrés Villaveces
Keyword(s):  
Small Index Property ◽  
Homogeneous Models ◽  
Small Index

AFFINE COVERS OF LIE GEOMETRIES AND THE AMALGAMATION PROPERTY

2002 ◽  
Vol 85 (3) ◽  
pp. 513-563 ◽  
Author(s):  
AMBAR CHOWDHURY ◽  
BRADD HART ◽  
ŽELJKO SOKOLOVIĆ
Keyword(s):  
Amalgamation Property ◽  
Subject Classification ◽  
Lie Geometry ◽  
Small Index Property ◽  
Small Index

Certain classes of smoothly approximable structures — the class of affine covers of Lie geometries — are shown to have the amalgamation property. In particular, this shows that any affine cover of a Lie geometry has the small index property.2000 Mathematical Subject Classification: 03C45.

The small index property for quadratic spaces

2005 ◽  
Vol 46 (1) ◽  
pp. 182-187
Author(s):  
I. V. Chirkov ◽  
M. A. Shevelin
Keyword(s):  
Small Index Property ◽  
Small Index

scholarly journals Reconstruction of homogeneous relational structures

2007 ◽  
Vol 72 (3) ◽  
pp. 792-802 ◽  
Author(s):  
Silvia Barbina ◽  
Dugald Macpherson
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Group Structure ◽  
Baire Category ◽  
Abstract Group ◽  
Relational Structures ◽  
Small Index Property ◽  
Categorical Structure ◽  
Small Index

This paper contains a result on the reconstruction of certain homogeneous transitive ω-categorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair (in a slightly non-standard sense, coming from Baire category).Reconstruction results give conditions under which the abstract group structure of the automorphism group Aut() of an ω-categorical structure determines the topology on Aut(), and hence determines up to bi-interpretability, by [1]; they can also give conditions under which the abstract group Aut() determines the permutation group ⟨Aut (), ⟩. so determines up to bi-definability. One such condition has been identified by M. Rubin in [12], and it is related to the definability, in Aut(), of point stabilisers. If the condition holds, the structure is said to have a weak ∀∃ interpretation, and Aut() determines up to bi-interpretability or, in some cases, up to bi-definability.A better-known approach to reconstruction is via the ‘small index property’: an ω-categorical stucture has the small index property if any subgroup of Aut() of index less than is open. This guarantees that the abstract group structure of Aut() determines the topology, so if is ω-categorical with Aut() ≅ Aut() then and are bi-interpretable.

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