Editing Boolean algebra (structure) (section)

== Generalizations ==
{{Algebraic structures|Lattice}}
Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a [[distributive lattice]] {{math|1=''B''}} is a generalized Boolean lattice, if it has a smallest element {{math|1=0}} and for any elements {{math|1=''a''}} and {{math|1=''b''}} in {{math|1=''B''}} such that {{math|1=''a'' ≤ ''b''}}, there exists an element {{math|1=''x''}} such that {{math|1=''a'' ∧ ''x'' = 0}} and {{math|1=''a'' ∨ ''x'' = ''b''}}. Defining {{math|1=''a'' \ ''b''}} as the unique {{math|1=''x''}} such that {{math|1=(''a'' ∧ ''b'') ∨ ''x'' = ''a''}} and {{math|1=(''a'' ∧ ''b'') ∧ ''x'' = 0}}, we say that the structure {{math|(''B'', ∧, ∨, \, 0)}} is a ''generalized Boolean algebra'', while {{math|(''B'', ∨, 0)}} is a ''generalized Boolean [[semilattice]]''. Generalized Boolean lattices are exactly the [[Ideal (order theory)|ideals]] of Boolean lattices.

A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an [[orthocomplemented lattice]]. Orthocomplemented lattices arise naturally in [[quantum logic]] as lattices of [[closed set|closed]] [[linear subspace]]s for [[separable space|separable]] [[Hilbert space]]s.