  We study countable structures from the viewpoint of enumeration reducibility.
Since enumeration reducibility is based on only positive information, in this
setting it is natural to consider structures given by their positive atomic
diagram -- the computable join of all relations of the structure. Fixing a
structure $\mathcal{A}$, a natural class of relations in this setting are the
relations $R$ such that $R^{\hat{\mathcal{A}}}$ is enumeration reducible to the
positive atomic diagram of $\hat{\mathcal{A}}$ for every $\hat{
\mathcal{A}}\cong \mathcal{A}$ -- the relatively intrinsically positively
enumerable (r.i.p.e.) relations. We show that the r.i.p.e. relations are
exactly the relations that are definable by $\Sigma^p_1$ formulas, a subclass
of the infinitary $\Sigma^0_1$ formulas. We then introduce a new natural notion
of the jump of a structure and study its interaction with other notions of
jumps. At last we show that positively enumerable functors, a notion studied by
Csima, Rossegger, and Yu, are equivalent to a notion of interpretability using
$\Sigma^p_1$ formulas.
