Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 stability of $T$
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 $A\subseteq \mathbb{N}$.

25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat