Editing Galois connection (section)

=== Antitone Galois connection ===
The above definition is common in many applications today, and prominent in [[lattice (order)|lattice]] and [[domain theory]]. However the original notion in Galois theory is slightly different. In this alternative definition, a Galois connection is a pair of ''antitone'', i.e. order-reversing, functions {{math|''F'' : ''A'' → ''B''}} and {{math|''G'' : ''B'' → ''A''}} between two posets {{mvar|A}} and {{mvar|B}}, such that

:{{math|''b'' ≤ ''F''(''a'')}} if and only if {{math|''a'' ≤ ''G''(''b'')}}.

The symmetry of {{mvar|F}} and {{mvar|G}} in this version erases the distinction between upper and lower, and the two functions are then called '''polarities''' rather than adjoints.{{sfn|Galatos|Jipsen|Kowalski|Ono|2007|p= 145}} Each polarity uniquely determines the other, since

:{{math|''F''(''a'')}} is the largest element {{mvar|b}} with {{math|''a'' ≤ ''G''(''b'')}}, and 
:{{math|''G''(''b'')}} is the largest element {{mvar|a}} with {{math|''b'' ≤ ''F''(''a'')}}.

The compositions {{math|''GF'' : ''A'' → ''A''}} and {{math|''FG'' : ''B'' → ''B''}} are the associated closure operators; they are monotone idempotent maps with the property {{math|''a'' ≤ ''GF''(''a'')}} for all {{mvar|a}} in {{mvar|A}} and {{math|''b'' ≤ ''FG''(''b'')}} for all {{mvar|b}} in {{mvar|B}}.

The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between {{mvar|A}} and {{mvar|B}} is just a monotone Galois connection between {{mvar|A}} and the [[duality (order theory)|order dual]] {{math|''B''<sup>op</sup>}} of {{mvar|B}}. All of the below statements on Galois connections can thus easily be converted into statements about antitone Galois connections.