Supersimple structures with a dense independent subset

TL;DR

This paper investigates expansions of supersimple theories with a dense independent subset, showing they have a unique theory, are supersimple, and exhibit a modular geometry of generics, extending prior o-minimal and geometric results.

Contribution

It introduces and analyzes $H$-structures in supersimple theories, establishing their theory's uniqueness, supersimplicity, and geometric properties, generalizing previous o-minimal and geometric findings.

Findings

All such expansions share the same theory.

Under certain conditions, saturated models are also $H$-structures.

The expansion is supersimple with a modular geometry of generics.

Abstract

Based on the work done in \cite{BV-Tind,DMS} 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 $\mathcal{G}(x)$, 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 $\omega^\alpha$ and the $SU$-rank is continuous, we take $\mathcal{G}(x)$ to be the type of elements of $SU$-rank $\omega^\alpha$ and we describe a natural "geometry of generics modulo $H$" associated with such expansions and show it is modular.