A partial equivalence relation is a binary relation satisfying the symmetry and transitivity conditions of an equivalence relation, but not necessarily the reflexivity condition. That is, a partial equivalence relation on is a binary relation such that for all and in , implies , and for all , , and in , and together imply .
Just as unary relations on a set correspond to subsets of and equivalence relations on correspond to quotients of , so partial equivalence relations on correspond to subquotients of . That is, the elements satisfying comprise a subset of , on which the relation restricts to a total equivalence relation specifying a further quotient.
Then this defines a partial equivalence relation on ; the corresponding subquotient of is the set of Cauchy real numbers. Normally, this definition of real number is split into two parts: those sequences satsifying the reflexivity condition of are the Cauchy sequences of rational numbers (under the absolute-value metric), and then we impose a total equivalence relation on the Cauchy sequences. But a single partial equivalence relation does all of the work. (This example generalises in the usual ways.)