Definition ($i$-sections). For $A\subset P(X)$ and $1\le i\le n$, the $i$-sections of $A$ are the families $A_{-}^{(i)},A_{+}^{(i)}\subset P(X-i)$ given by:

$$\begin{array}{llll}\hfill A_{-}^{(i)}& =\{x\in A:i\notin x\}& \hfill & \\ \hfill A_{+}^{(i)}& =\{x-i:x\in A,i\in x\}& \hfill & \end{array}$$

The $i$-compression of $A$ in the family $C_i(A)\subset P(X)$ given by:

• ${(C_i(A))}_{-}^{(i)}$ is the first $|A_{-}^{(i)}|$ elements of simplicial ordering on $P(X-i)$

• ${(C_i(A))}_{+}^{(i)}$ is the first $|A_{+}^{(i)}|$ elements of simplicial ordering on $P(X-i)$