Enriching a predicate and tame expansions of the integers

Given a structure $\mathcal{M}$ and a stably embedded $\emptyset$-definable set $Q$, we prove tameness preservation results when enriching the induced structure on $Q$ by some further structure $\mathcal{Q}$. In particular, we show that if $T=\text{Th}(\mathcal{M})$ and $\text{Th}(\mathcal{Q})$ are stable (resp., superstable, $\omega$-stable), then so is the theory $T[\mathcal{Q}]$ of the enrichment of $\mathcal{M}$ by $\mathcal{Q}$. 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 NSOP$_1$ pass from $\text{Th}(\mathcal{Q})$ to $T[\mathcal{Q}]$. 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 $(\mathbb{Z},+)$. More generally, we show that any stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) countable graph can be defined in a stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) expansion of $(\mathbb{Z},+)$ by some unary predicate $A\subseteq\mathbb{N}$.