Preordered class

In mathematics, a preordered class is a class with a preorder.

Definition

When dealing with a class C, it is possible to define a class relation on C as a subclass of the power class C² . Usually, the language of relations on a set is adopted.

A preordered class is a class with a preorder on it and an partially ordered class is a class with a partial order on it. Of course, any preordered set is a preordered class and any poset is a partially ordered class.

Equivalently, a preordered class is a thin category, that is, a category with at most one morphism from an object to another.

Examples

• In any category, the relation 'there exists a monomorphism from X to Y' is a preorder.
• The class Ord of ordinals is an ordered class with the order resulting from the preorder on Set described above.

References

• Nicola Gambino and Peter Schuster, Spatiality for formal topologies
• Adámek, Jiří (1990). Abstract and Concrete Categories (PDF). John Wiley & Sons. ISBN 0-471-60922-6. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)