attach a simplicial cell ball in (II) to change the shape of the boundary; Finally, we make a simplicial cell ball with the desired $h$-vector by using (I) again. The first technique (I) appeared in [@Ko]. Indeed, if $\partial {\mathbf{h}}$ is positive or $\sum_{k=0}^d h_k$ is odd, then the first technique is enough to prove the desired statement. However, we need (II) when $\partial {\mathbf{h}}$ has a zero entry and $\sum_{k=0}^d h_k$ is even.

Before proving the main lemmas, we first study what happens to the $h$-vector when we glue two simplicial posets. Two finite posets $P$ and $Q$ are said to be *isomorphic* if there is a bijection $f:P \to Q$, called an *isomorphism*, such that, for all $ \sigma, \tau \in P$, $\sigma> \tau$ if and only if $f(\sigma) > f(\tau)$. We write $P \cong Q$ if $P$ and $Q$ are isomorphic.

Let $P$ and $Q$ be simplicial posets. Let $I \subset P$ and $J \subset Q$ be order ideals which are isomorphic as posets. For a given isomorphism $f : I \to J$, we define the equivalence relation $\sim_f$ on $P \cup Q$ by $\sigma \sim_f f(\sigma)$ for all $ \sigma \in I$, and define $$P \cup_f  