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One-dimensional groups over an o-minimal structure

Vladimir Razenj

Annals of Pure and Applied Logic 53 (3):269-277 (1991)

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Intersection theory for o-minimal manifolds.Alessandro Berarducci & Margarita Otero - 2001 - Annals of Pure and Applied Logic 107 (1-3):87-119.details We develop an intersection theory for definable Cp-manifolds in an o-minimal expansion of a real closed field and we prove the invariance of the intersection numbers under definable Cp-homotopies . In particular we define the intersection number of two definable submanifolds of complementary dimensions, the Brouwer degree and the winding numbers. We illustrate the theory by deriving in the o-minimal context the Brouwer fixed point theorem, the Jordan-Brouwer separation theorem and the invariance of the Lefschetz numbers under definable Cp-homotopies. A. (...) Download Export citation Bookmark 7 citations
A descending chain condition for groups definable in o -minimal structures.Alessandro Berarducci, Margarita Otero, Yaa’cov Peterzil & Anand Pillay - 2005 - Annals of Pure and Applied Logic 134 (2):303-313.details We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest type-definable subgroup G00 of bounded index and G/G00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author. Download Export citation Bookmark 9 citations
Note on 'Normalisation for Bilateral Classical Logic with some Philosophical Remarks'.Nils Kürbis - 2021 - Journal of Applied Logics 7 (8):2259-2261.details This brief note corrects an error in one of the reduction steps in my paper 'Normalisation for Bilateral Classical Logic with some Philosophical Remarks' published in the Journal of Applied Logics 8/2 (2021): 531-556. Download Export citation Bookmark 2 citations
Connected components of definable groups, and o-minimality II.Annalisa Conversano & Anand Pillay - 2015 - Annals of Pure and Applied Logic 166 (7-8):836-849.details Download Export citation Bookmark 2 citations
Type-definability, compact lie groups, and o-minimality.Anand Pillay - 2004 - Journal of Mathematical Logic 4 (02):147-162.details We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures (...) Download Export citation Bookmark 23 citations
One-basedness and groups of the form G/G00.Davide Penazzi - 2011 - Archive for Mathematical Logic 50 (7-8):743-758.details We initiate a geometric stability study of groups of the form G/G00, where G is a 1-dimensional definably compact, definably connected, definable group in a real closed field M. We consider an enriched structure M′ with a predicate for G00 and check 1-basedness or non-1-basedness for G/G00, where G is an additive truncation of M, a multiplicative truncation of M, SO2(M) or one of its truncations; such groups G/G00 are now interpretable in M′. We prove that the only 1-based groups (...) Download Export citation Bookmark
O-Minimal Spectra, Infinitesimal Subgroups and Cohomology.Alessandro Berarducci - 2007 - Journal of Symbolic Logic 72 (4):1177 - 1193.details By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal "infinitesimal subgroup" G00 such that the quotient G/G00, equipped with the "logic topology", is a compact (real) Lie group. Our first result is that the functor G → G/G00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum (...) Download Export citation Bookmark 3 citations
Groups of dimension two and three over o-minimal structures.A. Nesin, A. Pillay & V. Razenj - 1991 - Annals of Pure and Applied Logic 53 (3):279-296.details Let G be a group definable in an o-minimal structure M. In this paper we show: Theorem. If G is a two-dimensional definably connected nonabelian group, then G is centerless and G is isomorphic to R+R*>0, for some real closed field R. Theorem. If G is a three-dimensional nonsolvable, centerless, definably connected group, then either G SO3 or G PSL2, for some real closed field R. Download Export citation Bookmark 5 citations
On variants of o-minimality.Dugald Macpherson & Charles Steinhorn - 1996 - Annals of Pure and Applied Logic 79 (2):165-209.details Download Export citation Bookmark 30 citations
Superrosy dependent groups having finitely satisfiable generics.Clifton Ealy, Krzysztof Krupiński & Anand Pillay - 2008 - Annals of Pure and Applied Logic 151 (1):1-21.details We develop a basic theory of rosy groups and we study groups of small Uþ-rank satisfying NIP and having finitely satisfiable generics: Uþ-rank 1 implies that the group is abelian-by-finite, Uþ-rank 2 implies that the group is solvable-by-finite, Uþ-rank 2, and not being nilpotent-by-finite implies the existence of an interpretable algebraically closed field. Download Export citation Bookmark 9 citations
Definable group extensions in semi‐bounded o‐minimal structures.Mário J. Edmundo & Pantelis E. Eleftheriou - 2009 - Mathematical Logic Quarterly 55 (6):598-604.details In this note we show: Let R = 〈R, <, +, 0, …〉 be a semi-bounded o-minimal expansion of an ordered group, and G a group definable in R of linear dimension m . Then G is a definable extension of a bounded definable group B by 〈Rm, +〉. Download Export citation Bookmark 3 citations

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