---- > [!definition] Definition. ([[cone and boundary of an element in a poset]]) > Let $(S, \leq)$ be a [[poset]]. The **cone** of $s \in S$ is the [[upper set|lower set]] $C(s)=S_{\leq s}=\{ t \in s : t \leq s \}$. > The **boundary** of $s \in S$ is the [[upper set|lower set]] $\partial C(s)=S_{<s}=\{ t \in S: t < s \}$. Notice that $S_{\leq s}$ is obtained from $S_{<s}$ by appending the 'apex' $s$. ^definition > [!basicexample] > If $\mathcal{K}$ is an [[(abstract) simplicial complex]] and $S=\text{Cells}(\mathcal{K})$ its corresponding [[poset]], then the cone $C(F)$ of a face $F \in \mathcal{K}$ is the poset of faces (subsets) of the [[simplex]] $I$. The boundary of $F \in \mathcal{K}$ is the poset of *proper* faces of $F$. ^basic-example ---- #### ---- #### 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 > ```