In
mathematics, a preordered class is a
class equipped with a
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
.
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 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
*Public ...
on a set.
A preordered class is a class with a
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
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. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special ca ...
,
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) 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 binar ...
and
totally ordered set
In mathematics, a total 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 ( reflexiv ...
. 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 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 mathematics, is mostly concer ...
are no longer possible in this framework.
Equivalently, a preordered class is a
thin category, that is, a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
with at most one morphism from an object to another.
Examples
*In any
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
''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