mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a preordered class is a

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

equipped with a

preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. The name is meant to suggest that preorders are ''almost'' partial orders, ...

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

\times

C'' . Then, it is convenient to use the language of

relations Relation or relations may refer to: General uses * International relations, the study of interconnection of politics, economics, and law on a global level * Interpersonal relationship, association or acquaintance between two or more people * ...

on a set. A preordered class is a class with a

on it. ''Partially ordered class'' and ''totally ordered class'' are defined in a similar way. These concepts generalize respectively those of

preordered set In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name is meant to suggest that preorders are ''almost'' partial orders, but not quite, as they are not necessar ...

partially ordered set In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...

and

totally ordered set In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( ref ...

. However, it is difficult to work with them as in the ''small'' case because many constructions common in a

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

are no longer possible in this framework. Equivalently, a preordered class is a

thin category In mathematics, specifically category theory, a posetal category, or thin category, is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class (or a preordered set, if its objects for ...

, that is, a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

with at most one morphism from an object to another.

Examples

*In any

''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 *{{cite book , last = Adámek , first = Jiří , author2=Horst Herrlich , author3=George E. Strecker , year = 1990 , url = http://katmat.math.uni-bremen.de/acc/acc.pdf , title = Abstract and Concrete Categories , publisher = John Wiley & Sons , isbn = 0-471-60922-6 Order theory Set theory