simplicial poset $P$ (or a CW-complex $\Gamma(P)$) is called a *simplicial cell decomposition of a topological space $X$* if $\Gamma(P)$ is homeomorphic to $X$. It was proved in [@Mu] that face vectors of simplicial cell decompositions of some manifolds without boundary, such as product of spheres and real projective spaces, are characterized by conditions similar to those in Theorem \[1.1\]. It would be of interest to find manifolds $M$ with boundary such that face vectors of simplicial cell decompositions of $M$ can be characterized by conditions similar to those in Theorem \[1.2\].

This paper is organized as follows. In Section 2, we prove the necessity of Theorem \[1.2\] by using the face ring of a simplicial poset. In Section 3, we prove the sufficiency of Theorem \[1.2\] by using constructibility and shellability.

Proof of necessity
==================

In this section, we prove the necessity of Theorem \[1.2\]. To prove this, we need the face ring of a simplicial poset introduced by Stanley [@St]. We assume familiarity with commutative algebra theory.

We recall some basic notations on commutative algebra. Let $K$ be an infinite field, $R=R_0 \bigoplus R_1 \bigoplus \cdots$ a finitely generated commutative graded $K$-algebra with $R_0=K$, where $R_k$ is the 