This is from page 41 of "Abstract regular polytopes"
Suppose that we have a group $\Gamma = \langle\sigma_1,\cdots,\sigma_n\rangle$ generated by involutions and denote $\Gamma_J = \langle\sigma_j\ \vert\ j\notin J\rangle$ for $J\subset\{1,\cdots,n\}$. Assume that the generators satisfy the following property
$$\Gamma_I\cap\Gamma_J = \Gamma_{I\cup J}$$
Take the set $C_\Gamma := C(\Gamma;\sigma_1,\cdots,\sigma_n)$ as the set of right cosets $\Gamma_I\alpha$ whith $\alpha\in\Gamma$ and put on it the order $\Gamma_I\alpha\leqslant\Gamma_J\beta\iff\Gamma_I\alpha\subset\Gamma_J\beta$
Then thanks to the intersection property, $\Gamma_I\alpha\leqslant\Gamma_J\beta\iff I\subset J$ and $\alpha\beta^{-1}\in\Gamma_I$.
Why is this last assertion true?
In particular, I believe that it should be
$\Gamma_I\alpha\leqslant\Gamma_J\beta\iff I\supset J$ and $\alpha\beta^{-1}\in\Gamma_I$