Partial equivalence relation

In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) $R$ on a set $X$ is a relation that is symmetric and transitive. In other words, it holds for all $a, b, c \in X$ that:

if $a R b$ , then $b R a$ (symmetry)
if $a R b$ and $b R c$ , then $a R c$ (transitivity)

If $R$ is also reflexive, then $R$ is an equivalence relation.

Properties and applications

In a set-theoretic context, there is a simple structure to the general PER $R$ on $X$ : it is an equivalence relation on the subset $Y = \{ x \in X | x\,R\,x\} \subseteq X$ . ( $Y$ is the subset of $X$ such that in the complement of $Y$ ( $X\setminus Y$ ) no element is related by $R$ to any other.) By construction, $R$ is reflexive on $Y$ and therefore an equivalence relation on $Y$ . Notice that $R$ is actually only true on elements of $Y$ : if $x R y$ , then $y R x$ by symmetry, so $x R x$ and $y R y$ by transitivity. Conversely, given a subset Y of X, any equivalence relation on Y is automatically a PER on X.

PERs are therefore used mainly in computer science, type theory and constructive mathematics, particularly to define setoids, sometimes called partial setoids. The action of forming one from a type and a PER is analogous to the operations of subset and quotient in classical set-theoretic mathematics.

Every partial equivalence relation is a difunctional relation, but the converse doesn't hold.

The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence, i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reflexive.^[1]

Examples

A simple example of a PER that is not an equivalence relation is the empty relation $R=\emptyset$ (unless $X=\emptyset$ , in which case the empty relation is an equivalence relation (and is the only relation on $X$ )).

Kernels of partial functions

For another example of a PER, consider a set $A$ and a partial function $f$ that is defined on some elements of $A$ but not all. Then the relation $\approx$ defined by

x \approx y

if and only if

f

is defined at

x

f

is defined at

y

, and

f(x) = f(y)

is a partial equivalence relation but not an equivalence relation. It possesses the symmetry and transitivity properties, but it is not reflexive since if $f(x)$ is not defined then $x \not\approx x$ — in fact, for such an $x$ there is no $y \in A$ such that $x \approx y$ . (It follows immediately that the subset of $A$ for which $\approx$ is an equivalence relation is precisely the subset on which $f$ is defined.)

Functions respecting equivalence relations

Let X and Y be sets equipped with equivalence relations (or PERs) $\approx_X, \approx_Y$ . For $f,g : X \to Y$ , define $f \approx g$ to mean:

\forall x_0 \; x_1, \quad x_0 \approx_X x_1 \Rightarrow f(x_0) \approx_Y g(x_1)

then $f \approx f$ means that f induces a well-defined function of the quotients $X / \approx_X \; \to \; Y / \approx_Y$ . Thus, the PER $\approx$ captures both the idea of definedness on the quotients and of two functions inducing the same function on the quotient.

References

↑ J. Lambek (1996). "The Butterfly and the Serpent". In Aldo Ursini, Paulo Agliano. Logic and Algebra. CRC Press. pp. 161–180. ISBN 978-0-8247-9606-8.