Definable homomorphisms of abelian groups in o-minimal structures

Ya'acov Peterzil; Sergei Starchenko

doi:10.1016/s0168-0072(99)00016-0

Definable homomorphisms of abelian groups in o-minimal structures

Ya'acov Peterzil, Sergei Starchenko

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate the group ℋ of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure script N. We prove the existence of a "large", definable subgroup of ℋ. If ℋ contains an infinite definable set of homomorphisms then some definable subgroup of B (equivalently, a definable quotient of A) admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure script N but also in any structure definable in script N.

Original language	English
Pages (from-to)	1-27
Number of pages	27
Journal	Annals of Pure and Applied Logic
Volume	101
Issue number	1
DOIs	https://doi.org/10.1016/s0168-0072(99)00016-0
State	Published - 15 Nov 1999

Bibliographical note

Funding Information:
E-mail addresses: [email protected] (Y. Peterzil), [email protected] (S. Starchenko) 1The second author was partially supported by NSF.

Keywords

Definably compact groups
o-minimal structures

ASJC Scopus subject areas

Logic

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10.1016/s0168-0072(99)00016-0

Cite this

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abstract = "We investigate the group ℋ of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure script N. We prove the existence of a {"}large{"}, definable subgroup of ℋ. If ℋ contains an infinite definable set of homomorphisms then some definable subgroup of B (equivalently, a definable quotient of A) admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure script N but also in any structure definable in script N.",

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AU - Starchenko, Sergei

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N2 - We investigate the group ℋ of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure script N. We prove the existence of a "large", definable subgroup of ℋ. If ℋ contains an infinite definable set of homomorphisms then some definable subgroup of B (equivalently, a definable quotient of A) admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure script N but also in any structure definable in script N.

AB - We investigate the group ℋ of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure script N. We prove the existence of a "large", definable subgroup of ℋ. If ℋ contains an infinite definable set of homomorphisms then some definable subgroup of B (equivalently, a definable quotient of A) admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure script N but also in any structure definable in script N.

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Definable homomorphisms of abelian groups in o-minimal structures

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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