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

 if , then  (symmetry)
 if  and , then  (transitivity)

If  is also reflexive, then  is an equivalence relation.

Properties and applications 
If a relation  on a set  is a PER then  is an equivalence relation on the subset . However, given a set  and a subset , an equivalence relation on  need not be a PER on ; for instance, considering the set , the relation over  characterised by the set  is an equivalence relation on  but not a PER on  since it is neither symmetric nor transitive on .

Every partial equivalence relation is a difunctional relation, but the converse does not hold.

Each partial equivalence relation is a right Euclidean relation. The contrary does not hold: for example, xRy on natural numbers, defined by 0 ≤ x ≤ y+1 ≤ 2, is right Euclidean, but neither symmetric (since e.g. 2R1, but not 1R2) nor transitive (since e.g. 2R1 and 1R0, but not 2R0). Similarly, each partial equivalence relation is a left Euclidean relation, but not vice versa. Each partial equivalence relation is quasi-reflexive, as a consequence of being Euclidean.

In non-set-theory settings

In type theory, constructive mathematics and their applications to computer science, constructing analogues of subsets is often problematic—in these contexts PERs are therefore more commonly used, particularly to define setoids, sometimes called partial setoids. Forming a partial setoid from a type and a PER is analogous to forming subsets and quotients in classical set-theoretic mathematics.

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.

Examples

A simple example of a PER that is not an equivalence relation is the empty relation , if  is not empty.

Kernels of partial functions
If  is a partial function on a set , then the relation  defined by 
  if  is defined at ,  is defined at , and 
is a partial equivalence relation, since it is clearly symmetric and transitive.

If  is undefined on some elements, then  is not an equivalence relation. It is not reflexive since if  is not defined then  — in fact, for such an  there is no  such that .  It follows immediately that the largest subset of  on which  is an equivalence relation is precisely the subset on which  is defined.

Functions respecting equivalence relations
Let X and Y be sets equipped with equivalence relations (or PERs) . For , define  to mean:

 

then  means that f induces a well-defined function of the quotients . Thus, the PER  captures both the idea of definedness on the quotients and of two functions inducing the same function on the quotient.

Equality of IEEE floating point values
IEEE 754:2008 floating point standard defines an "EQ" relation for floating point values. This predicate is symmetrical and transitive, but is not reflexive because of the presence of NaN values that are not EQ to themselves.

Notes

References

 Mitchell, John C. Foundations of programming languages. MIT Press, 1996.
 D.S. Scott. "Data types as lattices". SIAM Journ. Comput., 3:523-587, 1976.

Symmetric relations
Transitive relations
Equivalence (mathematics)