Partial equivalence relation

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In mathematics, a partial equivalence relation (often abbreviated as PER) 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:

  1. if a R b, then b R a (symmetry)
  2. if a R b and b R c, then a R c (transitivity)

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

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.


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[edit]

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[edit]

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.


See also[edit]