Based on the work done in [][] in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking‐independent elements that is dense inside a partial type, which we call H‐structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again H‐structures. We prove that under these assumptions the expansion is supersimple and (...) characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one‐basedness and CM‐triviality. In the one‐based case, when T has SU‐rank and the SU‐rank is continuous, we take to be the type of elements of SU‐rank and we describe a natural “geometry of generics modulo H” associated with such expansions and show it is modular. (shrink)

Fix a weakly minimal (i.e. superstable U-rank 1) structure M. Let M∗ be an expansion by constants for an elementary substructure, and let A be an arbitrary subset of the universe M. We show that all formulas in the expansion (M∗,A) are equivalent to bounded formulas, and so (M,A) is stable (or NIP) if and only if the M-induced structure AM on A is stable (or NIP). We then restrict to the case that M is a pure abelian group with (...) a weakly minimal theory, and AM is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of (Z,+). Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form (M,A). Most notably, we show that if (G,+) is a weakly minimal additive subgroup of the algebraic numbers, A⊆G is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of A is a root of unity, then (G,+,B) is superstable for any B⊆A. (shrink)

We study perfectly trivial theories, 1-based theories, stable varieties, and their mutual interaction. We give a structure theorem for the models of a complete perfectly trivial stable theory without DOP: any model is the algebraic closure of a nonforking regular tree of elements. We also give a structure theorem for stable varieties, all of whose completions have NDOP. Such a variety is a varietal product of an affine variety and a combinatorial variety of an especially simple form.

We construct a stable one-based, trivial theory with a reduct which is not trivial. This answers a question of John B. Goode. Using this, we construct a stable theory which is n-ample for all natural numbers n, and does not interpret an infinite group.

Building on Pierre Simon’s notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rankmsuch that$1\leq m \leq \omega $. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality calledm-determinacy and show that theories of distality rankmrequire certain products to bem-determined. Furthermore, for NIP theories, this behavior characterizesm-distality. If we narrow the scope to stable theories, we observe thatm-distality can be (...) characterized by the maximum cycle size found in the forking “geometry,” so it coincides with$(m-1)$-triviality. On a broader scale, we see thatm-distality is a strengthening of Saharon Shelah’s notion ofm-dependence. (shrink)