Proposition 1.12 . [04RK]

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Proposition 1.12.

Let $\Pi$ be a dual $\Delta$ -complex and let $\Delta^{\prime}\subset\Delta$ be a $(k+1)$ -dimensional face. Then the intersection $\bar{\Pi}\cap\Delta^{\prime}$ is a compactification of a dual $\Delta^{\prime}$ -complex $\Pi^{\prime}$ . If $\Pi$ is maximal then $\Pi^{\prime}$ is also maximal.