Can a Relation Be Both a Partial Order and an Equivalence Relation?

Yes, a relation can be both a partial order and an equivalence relation. However, this typically implies that the relation possesses specific properties unique to both types of relations. Let's explore the definitions, properties, and examples of these relations to understand this concept more deeply.

Definitions of Partial Order and Equivalence Relation

Partial Order

A relation R on a set A is a partial order if it satisfies the following properties:

Reflexive: For all a in A, a R a. Antisymmetric: For all a, b in A, if a R b and b R a, then a b. Transitive: For all a, b, c in A, if a R b and b R c, then a R c.

Equivalence Relation

A relation R on a set A is an equivalence relation if it satisfies the following properties:

Reflexive: For all a in A, a R a. Symmetric: For all a, b in A, if a R b, then b R a. Transitive: For all a, b, c in A, if a R b and b R c, then a R c.

Key Points

Both relations are required to be reflexive and transitive. However, the primary difference is that a partial order is antisymmetric, while an equivalence relation is symmetric. This distinction makes it uncommon for a relation to be both a partial order and an equivalence relation.

Example: A Relation Can Be Both

A relation can be both a partial order and an equivalence relation if it is defined in a specific way. For instance, consider the following relation on a set A

Let R be the equality relation on A, denoted by a b. This relation is:

Reflexive: For all a in A, a a. Symmetric: For all a, b in A, if a b, then b a. Transitive: For all a, b, c in A, if a b and b c, then a c.

This relation is also a partial order if we consider it with the same properties:

Reflexive: For all a in A, a a. Antisymmetric: For all a, b in A, if a b and b a, then a b. Transitive: For all a, b, c in A, if a b and b c, then a c.

Thus, the equality relation is both a partial order and an equivalence relation.

Conclusion

The only way for a relation to be both a partial order and an equivalence relation is if it is trivial, specifically, it must relate every element to itself and no other elements. The most common example of such a relation is the equality relation on a set where all elements are considered equivalent.

While both partial orders and equivalence relations share the properties of reflexivity and transitivity, their differences in antisymmetry and symmetry make them distinct. Understanding these properties helps in identifying and applying the correct type of relation in various mathematical and real-world scenarios.