graded component of $R$ of degree $k$, and let $M$ be a finitely generated graded $R$-module. The *Hilbert series $H_M(t)$ of $M$* is the formal power series $H_M(t)=\sum _{k = 0}^\infty (\dim_K M_k )t^k$. The *Krull dimension* of $M$, denoted $\dim M$, is the minimal $m$ for which there exist homogeneous elements $\theta_1,\dots,\theta_m \in R$ of positive degrees such that $M/(\theta_1,\dots,\theta_m)M$ is a finite dimensional $K$-vector space. When $\dim M=d$, a sequence of homogeneous elements $\theta_1,\dots,\theta_d \in R$ of positive degrees such that $M/(\theta_1,\dots,\theta_d)M$ is a finite dimensional $K$-vector space is called a *homogeneous system of parameters* (*h.s.o.p. *for short) *of* $M$. Moreover, if $\theta_1,\dots,\theta_d$ have degree $1$, then we call $\theta_1,\dots,\theta_d$ a *linear system of parameters* (*l.s.o.p. *for short) *of* $M$. We say that $M$ is *Cohen-Macaulay* if, for every (equivalently some) h.s.o.p. $\theta_1,\dots,\theta_d$ of $M$, one has that $\theta_i$ is a non-zero divisor of $M/(\theta_1,\dots,\theta_{i-1})M$ for all $i=1,2,\dots,d$. Also, $R$ is said to be Cohen-Macaulay if it is a Cohen-Macaulay $R$-module.

Let $P$ be a $(d-1)$-dimensional simplicial poset and let $A=K[x_\sigma: \sigma \in P \setminus \{ \hat 0\}]$ be the polynomial ring over an infinite field $K$ in indeterminates indexed by the elements in $P \setminus \{\hat 0\}$. 