# Preordered class

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{{Short description|Class equipped with a preorder}}
In [mathematics](/source/mathematics), a '''preordered class''' is a [class](/source/class_(mathematics)) equipped with a [preorder](/source/preorder).

==Definition==
When dealing with a class ''C'', it is possible to define a class relation on ''C'' as a [subclass](/source/Subclass_(set_theory)) of the power class ''C <math> \times </math> C'' . Then, it is convenient to use the language of [relations](/source/relation_(mathematics)) on a set.

A '''preordered class''' is a class with a [preorder](/source/preorder) on it. ''Partially ordered class'' and ''totally ordered class'' are defined in a similar way. These concepts  generalize respectively those of [preordered set](/source/preordered_set), [partially ordered set](/source/partially_ordered_set) and [totally ordered set](/source/totally_ordered_set). However, it is difficult to work with them as in the ''small'' case because many constructions common in a [set theory](/source/set_theory) are no longer possible in this framework.

Equivalently, a preordered class is a '''[thin category](/source/thin_category)''', that is, a [category](/source/category_(mathematics)) with at most one morphism from an object to another.

==Examples==
*In any [category](/source/category_(mathematics)) ''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](/source/ordinal_number)s 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}}

Category:Order theory
Category:Set theory

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