Definition. Partial equivalence relation [math-L43K]

A partial equivalence relation (PER) on $\N$ is a subset $R \subseteq \N \times \N$ which, as a relation, is

1. symmetric
2. and transitive

An important property is quasi-reflexive, i.e. for $x, y \in \N$ and $xRy$, we have $xRx$ and $yRy$. Since we can combine $xRy$ and $yRx$ (symmetry) by transitivity to get $xRx$. For $yRy$ is similar.