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Concept
Partial equivalence relation
Summary
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.
Definition
Formally, a relation R on a set X is a PER if it holds for all a, b, c \in X that:
if a R b, then b R a (symmetry)
if a R b and b R c, then a R c (transitivity)
Another more intuitive definition is that R on a set X is a PER if there is some subset Y of X such that R \subseteq Y \times Y and R is an equivalence relation on Y. The two definitions are seen to be equivalent by taking Y = { x \in X \mid x,R,x}. Properties and applications The following propOfficial source
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