---- > [!definition] Definition. ([[poset]]) > A [[relation]] $\leq$ on a set $A$ is called **partial preorder** if > > 1. (Reflexivity) $x \leq x$ for all $x \in A$; > 2. (Transitivity) $(x \leq y \text{ and }y \leq z) \implies x \leq z$ for all $x,y,z \in A$; > > A pair $(A, \leq)$ consisting of a set $A$ and an preorder relation $\leq$ is called a **preposet** or **preordered set**. > > A **partially ordered set (poset)** is a preposet that additionally satisfies > > 3. (Antisymmetry) $(x \leq y \text{ and }y \leq x) \implies x=y$. > > If *moreover* a fourth condition > > 4. (Comparability) If $x,y \in A$ with $x \neq y$, then $x \leq y$ or $y \leq x$ > > holds, the poset $(A, \leq)$ is upgraded to be called a **totally ordered set**, or a **chain**. > > A given (pre)poset ($A, \leq$) may be viewed as a (small) [[category]], by declaring the elements of $A$ as objects and placing a single [[category|morphism]] $a_{1} \to a_{2}$ if $a_{1} \leq a_{2}$. An order-preserving map between posets is precisely a [[covariant functor|functor]] between their respective [[category|categories]]. > The consequent [[subcategory]] [[category of small categories|of]] $\mathsf{Cat}$ — that of (pre)posets and [[monotonic map|order-preserving]] maps — is denoted $\mathsf{PrePos}$ resp. $\mathsf{Pos}$. [[the category of preposets is equivalent to that of Alexandrov spaces|It is equivalent]] to the [[category]] $\mathsf{AlexTop}$ of [[Alexandrov topology|Alexandrov topological spaces]]. - [ ] there are more interesting categorical things to say, cf. posetal categories and lecture notes ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```