In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a preordered class is a class equipped with a preorder

Preorder

In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...

.

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

Relation (mathematics)

In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...

 on a set.

A preordered class is a class with a preorder

Preorder

In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...

 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

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

 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

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

 are no longer possible in this framework.

Equivalently, a preordered class is a thin category, that is, a category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 with at most one morphism from an object to another.

Examples

• In any category
  Category (mathematics)
  In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
  
   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 ordinal
  Ordinal number
  In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
  
  s is a totally ordered class with the classical ordering of ordinals.