Boolean lattice

From AoPSWiki

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.