In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, then the relation is an equivalence relation.
Formally, a relation on a set is a PER if it holds for all that:
- if , then (symmetry)
- if and , then (transitivity)
Properties and applications
The following properties hold for a partial equivalence relation on a set :
- is an equivalence relation on the subset .[note 1]
- difunctional: the relation is the set for two partial functions and some indicator set
- right and left Euclidean: For , and implies and similarly for left Euclideanness and imply
- quasi-reflexive: If and , then and .[note 2]
None of these properties is sufficient to imply that the relation is a PER.[note 3]
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.
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
The 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.
- By construction, is reflexive on and therefore an equivalence relation on .
- This follows since if , then by symmetry, so and by transitivity. It is also a consequence of the Euclidean properties.
- For the equivalence relation, consider the set and the relation . is an equivalence relation on but not a PER on since it is neither symmetric (, but not ) nor transitive ( and , but not ). For Euclideanness, 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).
- Scott, Dana (September 1976). "Data Types as Lattices". SIAM Journal on Computing. 5 (3): 560. doi:10.1137/0205037.
- Mitchell, John C. (1996). Foundations for programming languages. Cambridge, Mass.: MIT Press. pp. 364–365. ISBN 0585037892.
- Encyclopaedia Britannica (EB); although EB's notion of quasi-reflexivity is Wikipedia's notion of left quasi-reflexivity, they coincide for symmetric relations.
- Salveson, A.; Smith, J.M. (1988). "The strength of the subset type in Martin-Lof's type theory".  Proceedings. Third Annual Information Symposium on Logic in Computer Science. pp. 384–391. doi:10.1109/LICS.1988.5135. ISBN 0-8186-0853-6. S2CID 15822016.
- 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.