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.
Bibliographical noteFunding Information:
E-mail addresses: email@example.com (Y. Peterzil), firstname.lastname@example.org (S. Starchenko) 1The second author was partially supported by NSF.
- Definably compact groups
- o-minimal structures
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