Boolean lattice

Given any set $S$, the boolean lattice $B(S)$ is a partially ordered set whose elements are those of $\mathcal{P}(S)$, the power set of $S$, ordered by inclusion ($\subset$).

When $S$ has a finite number of elements (say $|S| = n$), the boolean lattice associated with $S$ is usually denoted $B_n$. Thus, the set $S = \{1, 2, 3\}$ is associated with the boolean lattice $B_3$ with elements $\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}$ and $\{1, 2, 3\}$, among which $\emptyset$ is smaller than all others, $S = \{1, 2, 3\}$ is larger than all others, and the other six elements satisfy the relations $\{1\}, \{2\} \subset \{1,2\}$, $\{1\}, \{3\} \subset \{1,3\}$, $\{2\}, \{3\} \subset \{2,3\}$ and no others.

The Hasse diagram for $B_3$ is given below:

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Every boolean lattice is a distributive lattice, and the poset operations meet and join correspond to the set operations union and intersection.

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