Then the next lemma follows from Lemma \[3.1\].

\[3.2\] With the same notation as in Lemma \[3.1\], if $I \cong \Delta_d(k)$ then $$h(P \cup_I Q)=h(P) + h(Q) - {\mathbf{e}}_0 + {\mathbf{e}}_{k}.$$

To apply Lemmas \[3.1\] and \[3.2\] efficiently, it is convenient to have a nice way to construct simplicial cell balls by gluing two simplicial cell balls. Constructibility and shellability give such nice ways.

A $(d-1)$-dimensional simplicial poset $P$ is said to be *constructible* if

-   $P$ is a Boolean algebra of rank $d$, or

-   there are $(d-1)$-dimensional constructible simplicial posets $Q$ and $Q'$ such that $P=Q { {}_I \cup_{I'} } Q'$ for some order ideals $I \subset Q$ and $I' \subset Q'$ and that $I \cong I'$ is a $(d-2)$-dimensional constructible simplicial poset.

A simplicial poset is said to be *pure* if all its facets have the same rank. A $(d-1)$-dimensional pure simplicial poset $P$ is said to be *shellable* if there is an order $\sigma_1,\sigma_2,\dots,\sigma_r$ of the facets of $P$, called a *shelling of $P$*, such that, for $i=2,3,\dots,r$, $\langle \sigma_1,\sigma_2,\dots,\sigma_{i-1} \rangle \cap \langle \sigma_i \rangle$ is isomorphic to $\Delta_d(k_i)$ for some $1 \leq k_i \leq d$.

Note that shellable simplicial posets are constructible since $\Delta_d(k)$ is 