We define the grading of $A$ by $\deg x_\sigma = {\ensuremath{\mathrm{rank}}\hspace{1pt}}\sigma$. The *face ring of $P$* is the quotient ring $K[P]=A/I_P$, where $I_P$ is the ideal generated by the following elements:

-   $x_\sigma x_\tau$, if $\sigma,\tau \in P$ have no common upper bounds in $P$.

-   $x_\sigma x_\tau - x_{\sigma \wedge \tau} \sum_\rho x_\rho$, where the summation runs over the all minimal upper bounds of $\sigma$ and $\tau$ and where $\sigma \wedge \tau$ is the meet (largest lower bound) of $\sigma$ and $\tau$, otherwise. (We consider $x_{\sigma \wedge \tau}=1$ if $\sigma \wedge \tau=\hat 0$.)

Note that, for all $\sigma,\tau \in P$ such that $\sigma$ and $\tau$ have a common upper bound $\rho$, there is the unique largest lower bound of $\sigma$ and $\tau$ since $[\hat 0,\rho]$ is a Boolean algebra. Also, since generators of $I_P$ are homogeneous, the ring $K[P]$ is graded. In the special case when $P$ is the face poset of a simplicial complex, the ring $K[P]$ is isomorphic to the face ring (Stanley-Reisner ring) of a simplicial complex [@St2 p. 53]. See [@MMP p. 467].

A simplicial poset $P$ is said to be *Cohen-Macaulay* if the ring $K[P]$ is Cohen-Macaulay. Note that this is equivalent to saying 