empty set (considered as a preordered set) is the initial object of PreOrd, and the terminal objects are precisely the singleton preordered sets. There...

category of sets, whose objects are sets and whose arrows are functions. Category theory is a branch of mathematics that seeks to generalize all of mathematics...

order – preorder that is antisymmetric and total Directed set Category of preordered sets Prewellordering Well-quasi-ordering For "proset", see e.g....

subcategory consisting of Alexandrov-discrete spaces. Let Preord denote the category of preordered sets consisting of preordered sets with order preserving...

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...

spaces Category of sets and relations Category of topological spaces Category of metric spaces Category of preordered sets Category of groups Category of abelian...

called topological sorting. Every poset (and every preordered set) may be considered as a category where, for objects x {\displaystyle x} and y , {\displaystyle...

Chicago, US Ord, the category of preordered sets Ord, the proper class of all ordinal numbers ord(V), the order type of a well-ordered set V ordn(a), the multiplicative...

representation categories of linear algebraic groups. A preordered monoid is a monoidal category in which for every two objects c , c ′ ∈ O b ( C ) {\displaystyle...

In the language of category theory, we then have a functor from the category of topological spaces to the category of preordered sets that assigns a topological...

amounts to a preordered class (or a preordered set, if its objects form a set). As suggested by the name, the further requirement that the category be skeletal...

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general...

a (2, 1)-category. The category Ord of preordered sets is a 2-category since each hom-set has a natural preordered structure; thus a category structure...

Every preordered set is an upper set of itself. The intersection and the union of any family of upper sets is again an upper set. The complement of any...

(X,\leq )} is a preordered set and ( X , F ) {\displaystyle (X,{\mathcal {F}})} is a field of sets. Like the topological fields of sets, preorder fields...

A preordered field is a field equipped with a preordering P . {\displaystyle P.} Its non-zero elements P ∗ {\displaystyle P^{*}} form a subgroup of the...

element of a directed preordered set is a greatest element. Indeed, a directed preordered set is characterized by equality of the (possibly empty) sets of maximal...

equivalence of the pro-category of the category of finite preordered sets, Pro ⁡ ( PoSet fin ) {\displaystyle \operatorname {Pro} (\operatorname {PoSet} ^{\text{fin}})}...

in an interior algebra of the form B(X) for some preordered set X giving the above-mentioned representation as a field of sets (a preorder field). This...

{\displaystyle \,\leq \,} on the set X , {\displaystyle X,} the pair ( X , ≤ ) {\displaystyle (X,\leq )} is called a preordered vector space and we say that...

infrequent). A map h : A → I {\displaystyle h:A\to I} between two preordered sets is order–preserving if whenever a , b ∈ A  satisfy  a ≤ b ,  then ...

defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection"...

In mathematics, a subset B ⊆ A {\displaystyle B\subseteq A} of a preordered set ( A , ≤ ) {\displaystyle (A,\leq )} is said to be cofinal or frequent...

{\displaystyle b} of S , {\displaystyle S,} a ≤ b {\displaystyle a\leq b} (resp. a ≥ b {\displaystyle a\geq b} ). A preordered vector lattice is a preordered vector...

y'(x)=\left(x\mapsto -{\sqrt {2}}\right)'=0} . For functions between preordered sets, constant functions are both order-preserving and order-reversing;...

finitely many open sets (and only finitely many closed sets). A topology on a finite set can also be thought of as a sublattice of ( P ( X ) , ⊂ ) {\displaystyle...

arXiv:math/0111048. Krishnan, Sanjeevi (2009). "A Convenient Category of Locally Preordered Spaces". Applied Categorical Structures. 17 (5): 445–466. arXiv:0709...

\varnothing } is a set and for every a ∈ A , {\displaystyle a\in A,} ( I a , ≤ ) {\displaystyle \left(I_{a},\leq \right)} is a preordered set. Then the product...

contains upper bounds of finite subsets, but does not have to be a lower set. Furthermore, it is often generalized to preordered sets. A subset which is...

branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the...