Partial equivalence relations

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 $R\left(x,y\right)\subseteq {X}^{2}$ such that for all $x$ and $y$ in $X$, $R\left(x,y\right)$ implies $R\left(y,x\right)$, and for all $x$, $y$, and $z$ in $X$, $R\left(x,y\right)$ and $R\left(y,z\right)$ together imply $R\left(x,z\right)$.

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 $R\left(x,x\right)$ 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

$\underset{i,j\to \infty }{\mathrm{lim}}\mid {x}_{i}-{y}_{j}\mid =0.$\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.)

Revised on September 12, 2012 23:41:59 by Toby Bartels (98.19.47.153)