Editing Boolean algebra (structure) (section)

== Definition ==

A '''Boolean algebra''' is a [[Set (mathematics)|set]] {{math|1=''A''}}, equipped with two [[binary operation]]s {{math|1=∧}} (called "meet" or "and"), {{math|1=∨}} (called "join" or "or"), a [[unary operation]] {{math|1=¬}} (called "complement" or "not") and two elements {{math|1=0}} and {{math|1=1}} in {{math|1=''A''}} (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols {{math|1=⊥}} and {{math|1=⊤}}, respectively), such that for all elements {{math|''a''}}, {{math|''b''}} and {{math|''c''}} of {{math|''A''}}, the following [[axiom]]s hold:{{sfn|Davey|Priestley|1990|pp=109, 131, 144}}

:: {| cellpadding=5
|{{math|1=''a'' ∨ (''b'' ∨ ''c'') = (''a'' ∨ ''b'') ∨ ''c''}}
|{{math|1=''a'' ∧ (''b'' ∧ ''c'') = (''a'' ∧ ''b'') ∧ ''c''}}
| [[associativity]]
|-
|{{math|1=''a'' ∨ ''b'' = ''b'' ∨ ''a''}}
|{{math|1=''a'' ∧ ''b'' = ''b'' ∧ ''a''}}
| [[commutativity]]
|-
|{{math|1=''a'' ∨ (''a'' ∧ ''b'') = ''a''}}
|{{math|1=''a'' ∧ (''a'' ∨ ''b'') = ''a''}}
| [[Absorption law|absorption]]
|-
|{{math|1=''a'' ∨ 0 = ''a''}}
|{{math|1=''a'' ∧ 1 = ''a''}}
| [[identity element|identity]]
|-
|{{math|1=''a'' ∨ (''b'' ∧ ''c'') = (''a'' ∨ ''b'') ∧ (''a'' ∨ ''c'')&nbsp;&nbsp;}}
|{{math|1=''a'' ∧ (''b'' ∨ ''c'') = (''a'' ∧ ''b'') ∨ (''a'' ∧ ''c'')&nbsp;&nbsp;}}
| [[distributivity]]
|-
|{{math|1=''a'' ∨ ¬''a'' = 1}}
|{{math|1=''a'' ∧ ¬''a'' = 0}}
| [[complemented lattice|complements]]
|}
Note, however, that the absorption law and even the associativity law can be excluded from the set of axioms as they can be derived from the other axioms (see [[#Axiomatics|Proven properties]]).

A Boolean algebra with only one element is called a '''trivial Boolean algebra''' or a '''degenerate Boolean algebra'''. (In older works, some authors required {{math|0}} and {{math|1}} to be ''distinct'' elements in order to exclude this case.){{citation needed|date=July 2020}}

It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that
: {{math|1=''a'' = ''b'' ∧ ''a''}} &nbsp;&nbsp;&nbsp; if and only if &nbsp;&nbsp;&nbsp; {{math|1=''a'' ∨ ''b'' = ''b''}}.
The relation {{math|≤}} defined by {{math|''a'' ≤ ''b''}} if these equivalent conditions hold, is a [[partial order]] with least element 0 and greatest element 1. The meet {{math|1=''a'' ∧ ''b''}} and the join {{math|1=''a'' ∨ ''b''}} of two elements coincide with their [[infimum]] and [[supremum]], respectively, with respect to ≤.

The first four pairs of axioms constitute a definition of a [[bounded lattice]].

It follows from the first five pairs of axioms that any complement is unique.

The set of axioms is [[duality (order theory)|self-dual]] in the sense that if one exchanges {{math|1=∨}} with {{math|1=∧}} and {{math|0}} with {{math|1}} in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its '''dual'''.{{sfn|Goodstein|2012|p=21ff}}