Definable homomorphisms of abelian groups in o-minimal structures

Abstract

We investigate the group H of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure 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 N but also in any structure definable in N .