Abstract

Given a structure [Formula: see text] and a stably embedded [Formula: see text]-definable set Q, we prove tameness preservation results when enriching the induced structure on Q by some further structure [Formula: see text]. In particular, we show that if [Formula: see text] and [Formula: see text] are stable (respectively, superstable, [Formula: see text]-stable), then so is the theory [Formula: see text] of the enrichment of [Formula: see text] by [Formula: see text]. Assuming simplicity of T, elimination of hyperimaginaries and a further condition on Q related to the behavior of algebraic closure, we also show that simplicity and NSOP1 pass from [Formula: see text] to [Formula: see text]. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of [Formula: see text]. More generally, we show that any stable (respectively, superstable, simple, NIP, NTP2, NSOP1) countable graph can be defined in a stable (respectively, superstable, simple, NIP, NTP2, NSOP1) expansion of [Formula: see text] by some unary predicate [Formula: see text].