  The complexity of equivalence relations has received much attention in the
recent literature. The main tool for such endeavour is the following
reducibility: given equivalence relations $R$ and $S$ on natural numbers, $R$
is computably reducible to $S$ if there is a computable function $f \colon
\omega \to \omega$ that induces an injective map from $R$-equivalence classes
to $S$-equivalence classes. In order to compare the complexity of equivalence
relations which are computable, researchers considered also feasible variants
of computable reducibility, such as the polynomial-time reducibility. In this
work, we explore $\mathbf{Peq}$, the degree structure generated by primitive
recursive reducibility on punctual equivalence relations (i.e., primitive
recursive equivalence relations with domain $\omega$). In contrast with all
other known degree structures on equivalence relations, we show that
$\mathbf{Peq}$ has much more structure: e.g., we show that it is a dense
distributive lattice. On the other hand, we also offer evidence of the
intricacy of $\mathbf{Peq}$, proving, e.g., that the structure is neither rigid
nor homogeneous.
