Editing Boolean algebra (structure) (section)

{{short description|Algebraic structure modeling logical operations}}
{{for-multi|an introduction to the subject|Boolean algebra|an alternative presentation|Boolean algebras canonically defined}}
{{Use dmy dates|date=November 2020}}
In [[abstract algebra]], a '''Boolean algebra''' or '''Boolean lattice''' is a [[complemented lattice|complemented]] [[distributive lattice]]. This type of [[algebraic structure]] captures essential properties of both [[Set (mathematics)|set]] operations and [[logic]] operations. A Boolean algebra can be seen as a generalization of a [[power set]] algebra or a [[field of sets]], or its elements can be viewed as generalized [[truth value]]s. It is also a special case of a [[De Morgan algebra]] and a [[Kleene algebra (with involution)]].

Every Boolean algebra [[#Boolean rings|gives rise]] to a [[Boolean ring]], and vice versa, with [[ring (mathematics)|ring]] multiplication corresponding to [[logical conjunction|conjunction]] or [[meet (mathematics)|meet]] ∧, and ring addition to [[exclusive or|exclusive disjunction]] or [[symmetric difference]] (not [[logical disjunction|disjunction]] ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the [[axiom]]s and theorems of Boolean algebra express the symmetry of the theory described by the [[Duality principle (Boolean algebra)|duality principle]].{{sfn|Givant|Halmos|2009|p=20}}
[[Image:Hasse diagram of powerset of 3.svg|thumb|right|250px|Boolean lattice of subsets]]
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