Editing Boolean algebra (structure) (section)

== Homomorphisms and isomorphisms ==
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A ''[[homomorphism]]'' between two Boolean algebras {{math|''A''}} and {{math|''B''}} is a [[function (mathematics)|function]] {{math|''f'' : ''A'' → ''B''}} such that for all {{math|''a''}}, {{math|''b''}} in {{math|''A''}}:
: {{math|1=''f''(''a'' ∨ ''b'') = ''f''(''a'') ∨ ''f''(''b'')}},
: {{math|1=''f''(''a'' ∧ ''b'') = ''f''(''a'') ∧ ''f''(''b'')}},
: {{math|1=''f''(0) = 0}},
: {{math|1=''f''(1) = 1}}.

It then follows that {{math|1=''f''(¬''a'') = ¬''f''(''a'')}} for all {{math|''a''}} in {{math|''A''}}. The [[class (set theory)|class]] of all Boolean algebras, together with this notion of morphism, forms a [[full subcategory]] of the [[category theory|category]] of lattices.
<!--The constant function with ''f''(''a'') = 1 for all ''a'' in ''A'' satisfies the first, second, and fourth conditions but not the third (unless ''B'' is the degenerate singleton Boolean algebra with 0 = 1), so it is not a Boolean algebra homomorphism.-->

An ''isomorphism'' between two Boolean algebras {{math|''A''}} and {{math|''B''}} is a homomorphism {{math|''f'' : ''A'' → ''B''}} with an inverse homomorphism, that is, a homomorphism {{math|''g'' : ''B'' → ''A''}} such that the [[function composition|composition]] {{math|''g'' ∘ ''f'' : ''A'' → ''A''}} is the [[identity function]] on {{math|''A''}}, and the composition {{math|''f'' ∘ ''g'' : ''B'' → ''B''}} is the identity function on {{math|''B''}}.  A homomorphism of Boolean algebras is an isomorphism if and only if it is [[bijection|bijective]].