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The first three pairs of axioms above: associativity, commutativity and absorption, mean that (''A'', <math>\land</math>, <math>\lor</math>) is a [[lattice (order)|lattice]]. Thus a Boolean algebra can also be equivalently defined as a [[distributive lattice|distributive]] [[complemented lattice]].
From these [[axioms]], one can show that the smallest element 0, the largest element 1, and the complement ¬''a'' of any element ''a'' are uniquely determined. For all ''a'' and ''b'' in ''A'', the following [[identity (mathematics)|identities]] also follow:
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