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

In set theory, a relation on a set is a PER if, and only if, is an equivalence relation on the subset . By construction, is reflexive on and therefore an equivalence relation on . Actually, can hold only on elements of : if , then by symmetry, so and by transitivity, that is, . 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^{[note 1]} nor transitive^{[note 2]} 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. 2*R*1, but not 1*R*2) nor transitive (since e.g. 2*R*1 and 1*R*0, but not 2*R*0). Similarly, each partial equivalence relation is a left Euclidean relation, but not vice versa. Each partial equivalence relation is quasi-reflexive,^{[1]} 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^{[2]}—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.^{[3]}

## 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

**^**Encyclopaedia Britannica (EB); although EB's and Wikipedia's notions of quasi-reflexivity differ in general, they coincide for symmetric relations.**^**https://ieeexplore.ieee.org/document/5135/**^**J. Lambek (1996). "The Butterfly and the Serpent". In Aldo Ursini; Paulo Agliano (eds.).*Logic and Algebra*. CRC Press. pp. 161–180. ISBN 978-0-8247-9606-8.

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