Given any set , the boolean lattice is a partially ordered set whose elements are those of , the power set of , ordered by inclusion ().
When has a finite number of elements (say ), the boolean lattice associated with is usually denoted . Thus, the set is associated with the boolean lattice with elements and , among which is smaller than all others, is larger than all others, and the other six elements satisfy the relations , , and no others.
The Hasse diagram for 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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