nLab
partial equivalence relation

Definitions

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 $X$ is a binary relation $R(x,y)\subseteq X^2$ such that for all $x$ and $y$ in $X$, $R(x,y)$ implies $R(y,x)$, and for all $x$, $y$, and $z$ in $X$, $R(x,y)$ and $R(y,z)$ together imply $R(x,z)$.

Just as unary relations on a set $X$ correspond to subsets of $X$ and equivalence relations on $X$ correspond to quotients of $X$, so partial equivalence relations on $X$ correspond to subquotients of $X$. That is, the elements satisfying $R(x,x)$ comprise a subset of $X$, on which the relation restricts to a total equivalence relation specifying a further quotient.

Examples

Consider the set $X$ of all infinite sequences of rational numbers. Let such sequences $x$ and $y$ be related if

$$\underset{i,j\to \mathrm{\infty }}{\lim }\mid x_i-y_j\mid =0.$$\lim_{i,j\to \infty} {|x_i - y_j|} = 0 .

Then this defines a partial equivalence relation $R$ on $X$; the corresponding subquotient of $X$ 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 $R$ 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.)