On groups and fields definable in o-minimal structures

Pillay, Anand

doi:10.1016/0022-4049(88)90125-9

Cited by 173 publications

(208 citation statements)

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“…Under assumptions of Theorem 3.6, using the properties of the topology τ and the fact that finitely many translates of V cover G, one can easily show that every definable subset of G is a finite Boolean combination of τ -open definable subsets of G (or equivalently, a union of finitely many locally τ -closed sets). Consequently using §2 of [Pi1], as in §2 [Pi2], one can show that any definable subgroup of G is τ -closed.…”

Section: The Function F Y Is Continuous and Assumes Values Inmentioning

confidence: 99%

“…Pillay in [Pi2] adapts Hrushovski's unpublished proof of a special case of Weil's 'group chunk theorem' [W] to show that a group definable in an o-minimal structure can be definably equipped with a topology making it a topological group and a definable manifold. The topology in question on a large definable subset of the considered definable group coincides with the usual topology induced by ordering of the structure.…”

Section: Introductionmentioning

confidence: 99%

“…Among those topological structures we have several important classes of models satisfying various minimality conditions, namely: models of weakly o-minimal/C-minimal theories in which acl has the exchange property as well as P -minimal structures. Both [Pi2] and [Mo] extensively use the theory of generics.…”

Section: Introductionmentioning

confidence: 99%

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Groups, group actions and fields definable in first‐order topological structures

Wencel

2012

Mathematical Logic Qtrly

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Given a group (G, ·), G ⊆ M m , definable in a first order structure M = (M, . . .) equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V ⊆ G and define a new topology τ on G with which (G, ·) becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from M m . Likewise we topologize transitive group actions and fields definable in M. These results require a series of preparatory facts concerning dimension functions, some of which might be of independent interest. IntroductionA. Pillay in [Pi2] adapts Hrushovski's unpublished proof of a special case of Weil's 'group chunk theorem' [W] to show that a group definable in an o-minimal structure can be definably equipped with a topology making it a topological group and a definable manifold. The topology in question on a large definable subset of the considered definable group coincides with the usual topology induced by ordering of the structure. It follows that a field definable in an o-minimal structure can be definably made into a topological field.As observed by A. Mosley in [Mo], Pillay's method works for groups and fields definable in sufficiently saturated first order topological structures (introduced in [Pi1]) on which certain very natural conditions have been imposed, like for example the exchange property of the algebraic closure. Among those topological structures we have several important classes of models satisfying various minimality conditions, namely: models of weakly o-minimal/C-minimal theories in which acl has the exchange property as well as P -minimal structures. Both [Pi2] and [Mo] extensively use the theory of generics. This paper contains an alternative approach towards the problem of topologization of definable algebraic objects. Instead of imposing the exchange property on acl, we consider topological structures equipped with so called dimension function.Such a context is related to that of [Mo] but does not coincide with it. For example, it is easy too see that if M is a weakly o-minimal structure and for any N M, acl has the exchange property, then the usual topological dimension (dim) is a dimension function in the sense of [vdD1]. On the other hand, if dim is a dimension function in a weakly o-minimal structure M, then acl has the exchange property in M (see [We06] for details).The paper is organized as follows. In §1 we fix some notation and discuss the basic notions that are used throughout the paper, like dimension functions and first order topological structures. In §2 we consider the issue of covering a group G definable in a first order structure equipped with a (weak) dimension function by a finite number of its translates. We give effective bounds on the number of translates of a large definable subset of G needed to cover G.Some of the preparatory results proved in §2 are used in §3- §5 to investigate groups, group actions and fields definable in first order topological structures equipped with a dimensi...

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Section: The Function F Y Is Continuous and Assumes Values Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Groups, group actions and fields definable in first‐order topological structures

Wencel

2012

Mathematical Logic Qtrly

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“…Note that J , being abstractly isomorphic to the connected commutative compact Lie group G/G 00 , is divisible, and thus J is definably connected in the sense of having no proper definable subgroups of finite index. The general theory ( [31]) of equipping definable groups in ominimal structures with a definable group manifold structure applies to J . We conclude using Proposition 8.6(i) that J with its definable manifold topology is locally Euclidean, and thus (by the special case of Hilbert's 5th problem due to Pontryagin) is a connected commutative Lie group, whose Lie group dimension coincides with its semi-o-minimal dimension.…”

Section: Proof Of Claim (Iii)mentioning

confidence: 99%

On NIP and invariant measures

Hrushovski¹,

Pillay²

2011

J. Eur. Math. Soc.

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269

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Abstract. We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd(A), (ii) analogous statements for Keisler measures and definable groups, including the fact that G 000 = G 00 for G definably amenable, (iii) definitions, characterizations and properties of "generically stable" types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o-minimal expansions of real closed fields.

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“…The group topology on G is the t-topology from [Pi1]. The assumption of definable connectedness is at no loss of generality, again by [Pi1].…”

Section: And (Ii) a ⊆ S Is Open In The Quotient Topology If And Only mentioning

confidence: 99%

Compact domination for groups definable in linear o-minimal structures

Eleftheriou

2009

Arch. Math. Logic

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On groups and fields definable in o-minimal structures

Cited by 173 publications

References 7 publications

Groups, group actions and fields definable in first‐order topological structures

Groups, group actions and fields definable in first‐order topological structures

On NIP and invariant measures

Compact domination for groups definable in linear o-minimal structures

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