face poset of this regular cell decomposition.

We will need the following notion and result. For instance, see [@OM Definition 4.7.14].

A pure $n$-dimensional polytopal complex $\Delta$ is [*shellable*]{} if it is $0$-dimensional (and hence a collection of a finite number of points), or if there is a linear order of the facets $F_{1}, F_{2}, \ldots, F_{k}$ of $\Delta$, called a *shelling order*, such that:

-   The boundary complex of $F_{1}$ is shellable.

-   For $1 < j \leq k$ the intersection of $\overline{F_{j}}$ with the union of the closures of the previous facets is nonempty and is the beginning of a shelling of the $(n-1)$-dimensional boundary complex of $F_{j}$, that is, $$\overline{F_{j}} \cap 
    (\overline{F_{1}} \cup \overline{F_{2}} \cup \cdots \cup \overline{F_{j-1})}
    = 
    \overline{G_{1}} \cup \overline{G_{2}} \cup \cdots \cup \overline{G_{r}} ,$$ where $G_{1}, G_{2}, \ldots, G_{r}, \ldots, G_{t}$ is a shelling order of $\partial F_{j}$ and $r\geq 1$.

([@OM Theorem 4.3.3].) Let $B$ be a chamber in ${\mathscr{T}}$. Then any linear extension of the chamber poset with base chamber $B$ is a shelling order on the facets of $\Sigma({\mathscr{L}})$. \[thm\_shelling\_order\]

Finally, we recall the definition of the star of a face in ${\mathscr{L}}$.

Let $C\in{\mathscr{L}}$. The *star* of $C$ in ${\mathscr{L}}$ 