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The complexity of equivalence relations has received much attention in therecent literature. The main tool for such endeavour is the followingreducibility: 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 classesto $S$-equivalence classes. In order to compare the complexity of equivalencerelations which are computable, researchers considered also feasible variantsof computable reducibility, such as the polynomial-time reducibility. In thiswork, we explore $\mathbf{Peq}$, the degree structure generated by primitiverecursive reducibility on punctual equivalence relations (i.e., primitiverecursive equivalence relations with domain $\omega$). In contrast with allother known degree structures on equivalence relations, we show that$\mathbf{Peq}$ has much more structure: e.g., we show that it is a densedistributive lattice. On the other hand, we also offer evidence of theintricacy of $\mathbf{Peq}$, proving, e.g., that the structure is neither rigidnor homogeneous.

Punctual equivalence relations and their (punctual) complexity