Model Theory with Applications to Algebra and Analysis
Volume 2
Edited by ZoĆ© Chatzidakis
Edited by Dugald Macpherson
Edited by Anand Pillay
Edited by Alex Wilkie
Publisher: Cambridge University Press
Print Publication Year: 2008
Online Publication Date:August 2010
Online ISBN:9780511735219
Paperback ISBN:9780521709088
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511735219.006
Subjects: Logic, categories and Sets
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Introduction
Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable group manifold (see Section 2). This implies that when the group has the order type of the reals, we have a real Lie group. The main lines of research in the subject so far have been the following:
On interpretability, we have a clear view, with final results in Theorems 4.1 and 4.3 below. Lines of research (2) and (3) can be seen as a way of comparing definable groups with real Lie groups (see Section 2). The best results on the Euler characteristic are those of Theorems 6.3 and 6.5. The study of the torsion begins the study of the algebraic properties of definable groups, and the best result about the algebraic structure of the torsion subgroups is Theorem 5.9. On the other hand, the cases in which Pillay's conjectures are proved are stated in Theorems 8.3 and 8.8.
pp. i-iv
pp. v-viii
pp. ix-xii
pp. xiii-xvi
Conjugacy in groups of finite Morley rank : Read PDF
pp. 1-58
Permutation groups of finite Morley rank : Read PDF
pp. 59-124
A survey of asymptotic classes and measurable structures : Read PDF
pp. 125-160
Counting and dimensions : Read PDF
pp. 161-176
A survey on groups definable in o-minimal structures : Read PDF
pp. 177-206
Decision problems in Algebra and analogues of Hilbert's tenth problem : Read PDF
pp. 207-236
Hilbert's Tenth Problem for function fields of characteristic zero : Read PDF
pp. 237-254
First-order characterization of function field invariants over large fields : Read PDF
pp. 255-272
Nonnegative solvability of linear equations in ordered Abelian groups : Read PDF
pp. 273-314
Model theory for metric structures : Read PDF
pp. 315-427