# Preordered class

In mathematics, a **preordered class** is a class equipped 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 *C C* . Then, it is convenient to use the language of relations on a set.

A **preordered class** is a class with a preorder on it. *Partially ordered class* and *totally ordered class* are defined in a similar way. These concepts generalize respectively those of preordered set, partially ordered set and totally ordered set. However, it is difficult to work with them as in the *small* case because many constructions common in a set theory are no longer possible in this framework.

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
*C*, when*D*is a class of morphisms of*C*containing identities and closed under composition, the relation 'there exists a*D*-morphism from*X*to*Y'*is a preorder on the class of objects of*C*. - The class
**Ord**of all ordinals is a totally ordered class with the classical ordering of ordinals.

## References

- Nicola Gambino and Peter Schuster, Spatiality for formal topologies
- Adámek, Jiří; Horst Herrlich; George E. Strecker (1990).
*Abstract and Concrete Categories*(PDF). John Wiley & Sons. ISBN 0-471-60922-6.