We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

We characterize þ-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.

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We let R be an o-minimal expansion of a field, V a convex subring, and (R 0 , V 0 ) an elementary substructure of (R, V ). Our main result is that (R, V ) considered as a structure in a language containing constants for all elements of R 0 is model complete relative to quantifier elimination in R, provided that k R (the residue field with structure induced from R) is o-minimal. Along the way we show that o-minimality of k R implies that the sets definable in k R are the same as the sets definable in k with structure induced from (R, V ). We also give a criterion for a superstructure of (R, V ) being an elementary extension of (R, V ).