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 relation that is symmetric and transitive. In other words, it holds for all that:
- if , then (symmetry)
- if and , then (transitivity)
Properties and applicationsEdit
In a set-theoretic context, there is a simple structure to the general PER on : it is an equivalence relation on the subset . ( is the subset of such that in the complement of ( ) no element is related by to any other.) By construction, is reflexive on and therefore an equivalence relation on . Notice that is actually only true on elements of : if , then by symmetry, so and by transitivity. Conversely, given a subset Y of X, any equivalence relation on Y is automatically a PER on X. Hence, in set theory one typically studies the equivalence relation associated with a PER, rather than the PER itself.
But 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.
Every partial equivalence relation is a difunctional relation, but the converse does not 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.
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.
A simple example of a PER that is not an equivalence relation is the empty relation (unless , in which case the empty relation is an equivalence relation (and is the only relation on )).
Kernels of partial functionsEdit
For another example of a PER, consider a set and a partial function that is defined on some elements of but not all. Then the relation defined by
- if and only if is defined at , is defined at , and
is a partial equivalence relation but not an equivalence relation. It possesses the symmetry and transitivity properties, but 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 subset of for which is an equivalence relation is precisely the subset on which is defined.)
Functions respecting equivalence relationsEdit
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 valuesEdit
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.
- Mitchell, John C. Foundations of programming languages. MIT Press, 1996.
- D.S. Scott. "Data types as lattices". SIAM Journ. Comput., 3:523-587, 1976.