Boolean algebra (structure)

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In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a Boolean algebra or Boolean lattice is a <a href="/facts/Complemented_lattice/KpwmsACp">complemented</a> <a href="/facts/Distributive_lattice/97DFz1cu">distributive lattice</a>. This type of <a href="/facts/Algebraic_structure/TkuXca24">algebraic structure</a> captures essential properties of both <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> operations and <a href="/facts/Logic/8JnpJLtt">logic</a> operations. A Boolean algebra can be seen as a generalization of a <a href="/facts/Power_set/FVuf8PDD">power set</a> algebra or a <a href="/facts/Field_of_sets/4E3vQl4f">field of sets</a>, or its elements can be viewed as generalized <a href="/facts/Truth_value/SY4e2w8l">truth values</a>. It is also a special case of a <a href="/facts/De_Morgan_algebra/rLygz2hz">De Morgan algebra</a> and a <a href="/facts/Kleene_algebra_(with_involution)/rLygz2hz">Kleene algebra (with involution)</a>.
</p><p>Every Boolean algebra gives rise to a <a href="/facts/Boolean_ring/lgko4uqq">Boolean ring</a>, and vice versa, with <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> multiplication corresponding to <a href="/facts/Logical_conjunction/62uyyn1d">conjunction</a> or <a href="/facts/Meet_(mathematics)/cYcLhRIR">meet</a> ∧, and ring addition to <a href="/facts/Exclusive_or/FHyr7hGQ">exclusive disjunction</a> or <a href="/facts/Symmetric_difference/W2QST1Qr">symmetric difference</a> (not <a href="/facts/Logical_disjunction/jbI2EGlM">disjunction</a> ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the <a href="/facts/Axiom/mLjP14Mc">axioms</a> and theorems of Boolean algebra express the symmetry of the theory described by the <a href="/facts/Duality_principle_(Boolean_algebra)/VkCToAmg">duality principle</a>.
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Boolean algebra (structure) open-in-new

Boolean algebra (structure)