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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)$ $$\mathcal{D}(e') \subseteq \mathcal{D}(e)$$

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Consider the set $E=\mathbb N$, and let $D\subset 2^E$ be the set of sets $X_k=\{1\}\cup\{j\in E:j\geq k\}$, together with $E$ and $\varnothing$. $D$ is totally ordered under inclusion, so it is a distributive lattice. Now let $e=1$. What is $D(e)$?

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What is k here ? is this for all $k \in \mathbb{N}$ so $D$ consists of the sets {1,2,3,4,5,....}, {1,3,4,....}, {1,4,5,...}, and so on ? In that case $D(e)$ should be {1} $\notin$ D. – AnkurVijay Jun 22 '11 at 16:24

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