---- > [!definition] Definition. ([[graded poset]]) > A **grading**, or **rank function**, on a (finite) [[poset]] $S$ is a function $\text{rk}:S \to \mathbb{Z}$ satisfying the following: > 1. (Strict [[monotonic map|monotonicity]]) If $s_{1}<s_{2}$ then $\text{rk }s_{1} < \text{rk }s_{2}$; > 2. (Respects [[covering relation]]) If $s_{1}<s_{2}$ is a [[covering relation]], then $\text{rk }s_{2}=\text{rk }s_{1}+1$; > 3. If $s$ is minimal, then $\text{rk }s=0$. > We call the pair $(S, \text{rk})$ a **graded poset**. > > These properties turn out to determine a unique grading on $S$, should a grading exist at all. Specifically, if $S$ admits a grading $\text{rk}:S \to \mathbb{Z}$, $s \in S$, and $s_{1}<s_{2}<\dots<s_{k}<s$ > is a maximal chain descending from $s$, then the rank of $s$ is necessarily the length of this chain: $\text{rk }s=k$. (It follows that if such a definition is not [[well-defined]] irrespective of the choice of maximal chain, then no grading exists.) > > > > [!justification] Justifying Uniqueness. > Suppose $S$ admits a grading $\text{rk}: S \to \mathbb{Z}$. Let $s \in S$. If $s$ is minimal, then $\text{rk }s=0$. Otherwise, $s$ has a descending maximal chain $s_{1}<\dots<s_{k}<s$ for some $k \geq 1$. Since this chain is maximal, each comparison in it is a [[covering relation|covering]]; the result follows inductively. ^justification > [!basicexample] > The [[poset]] $\text{Cells}(\mathcal{K})$ of nonempty simplices of an [[(abstract) simplicial complex]] $\mathcal{K}$ is naturally graded by dimension: $\text{rk }F=\text{dim }F$. ^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 > ```