Definable homomorphisms of abelian groups in o-minimal structures Original Research Article

We investigate the group View the MathML source of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure View the MathML source. We prove the existence of a “large”, definable subgroup of View the MathML source. If View the MathML source 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 View the MathML source but also in any structure definable in View the MathML source.