let E be a set . Let $\mathcal{D} \subseteq 2^E$ be a distributive lattice with $\phi, E \in \mathcal{D}$.
For each $ e \in E$, define
$\mathcal{D}(e) = \cap \{ X | e \in X \in \mathcal{D}\}$
My Doubt:
1. Is it necessary that $\mathcal{D}(e) \in \mathcal{D}$ ?
2. Why is the following claim true ?
For any e' \in \mathcal{D}(e) \mathcal{D}(e') \subseteq \mathcal{D}(e)