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 applications
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.
PERs are therefore[why?] 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.
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.
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 )).
In the Euclidean plane, two lines m and n are parallel lines when m ∩ n = ∅. The symmetry of this relation is obvious and the transitivity can be proven in the Euclidean plane, thus Euclidean parallelism is a partial equivalence relation. Nevertheless, mathematicians developing affine geometry prefer the facility of an equivalence relation and therefore sometimes revise the definition of parallelism to allow a line to be parallel to itself, making the new relation of "affine parallelism" that is a reflexive relation.
Kernels of partial functions
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 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.
- Mitchell, John C. Foundations of programming languages. MIT Press, 1996.
- D.S. Scott. "Data types as lattices". SIAM Journ. Comput., 3:523-587, 1976.