Expansions of o-minimal structures by dense independent sets

Let MM be an o-minimal expansion of a densely ordered group and HH be a pairwise disjoint collection of dense subsets of M   such that ⋃H⋃H is definably independent in MM. We study the structure (M,(H)H∈H)(M,(H)H∈H). Positive results include that every open set definable in (M,(H)H∈H)(M,(H)H∈H) is definable in MM, the structure induced in (M,(H)H∈H)(M,(H)H∈H) on any H0∈HH0∈H is as simple as possible (in a sense that is made precise), and the theory of (M,(H)H∈H)(M,(H)H∈H) eliminates imaginaries and is strongly dependent and axiomatized over the theory of MM in the most obvious way. Negative results include that (M,(H)H∈H)(M,(H)H∈H) does not have definable Skolem functions and is neither atomic nor satisfies the exchange property. We also characterize (model-theoretic) algebraic closure and thorn forking in such structures. Throughout, we compare and contrast our results with the theory of dense pairs of o-minimal structures.