---- > [!definition] Definition. ([[strict order relation]]) > > > A [[relation]] lt;$ on a set $A$ is a called an **strict order relation** if it has the following properties: > > 1. (Nonreflexivity) For no $x$ in $A$ do we have $x < x$; > 2. (Transitivity) $x<y$ and $y<z$ imply $x<z.$ > A pair ($A, <$) consisting of a set $A$ and a strict order relation is called a **strict partially ordered set (strict poset)**. Should it be the case that a third condition > >3. (Comparability) If $x,y \in A$ with $x \neq y$, then $x < y$ or $y<x$ > holds, $(A, <)$ is upgraded to be called a **totally strictly ordered set**. ---- #### ---- #### 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 > ```