This talk concerns a couple properties of the theory obtained by adding a dense/codense algebraic substructure to an o-minimal expansion of an ordered divisible abelian group. I will discuss a characterization of when the expansion of an o-minimal group by a generic subgroup has a model companion. This characterization proves to be geometric in essence, and hence is similar in spirit to criteria for the property of near-model completeness. I will discuss a few examples of an o-minimal theory with a predicate for an algebraic substructure that is not generic, but satisfies some geometric criteria that imply near-model completeness. Namely, the examples are pairs of ordered vector spaces with different base fields, and pairs of fields such that one is real closed and one is pseudo-real closed.
