Editing Galois connection (section)

====Image and inverse image====
If {{math|&thinsp;''f'' : ''X'' → ''Y''}} is a [[function (mathematics)|function]], then for any subset {{mvar|M}} of {{mvar|X}} we can form the [[image (mathematics)|image]] {{math|''F''(''M''&hairsp;) {{=}} &thinsp;''f''&thinsp;''M'' {{=}} {&thinsp;''f''&thinsp;(''m'') {{!}} ''m'' ∈ ''M''} }} and for any subset {{mvar|N}} of {{mvar|Y}} we can form the [[inverse image]] {{math|''G''(''N''&hairsp;) {{=}} &thinsp;''f''&nbsp;<sup>−1</sup>''N'' {{=}} {''x'' ∈ ''X'' {{!}} &thinsp;''f''&thinsp;(''x'') ∈ ''N''}.}} Then {{mvar|F}} and {{mvar|G}} form a monotone Galois connection between the power set of {{mvar|X}} and the power set of {{mvar|Y}}, both ordered by inclusion ⊆. There is a further adjoint pair in this situation: for a subset {{mvar|M}} of {{mvar|X}}, define {{math|''H''(''M'') {{=}} {''y'' ∈ ''Y'' {{!}} &thinsp;''f''&nbsp;<sup>−1</sup>{''y''} ⊆ ''M''}.}} Then {{mvar|G}} and {{mvar|H}} form a monotone Galois connection between the power set of {{mvar|Y}} and the power set of {{mvar|X}}. In the first Galois connection, {{mvar|G}} is the upper adjoint, while in the second Galois connection it serves as the lower adjoint.

In the case of a [[quotient group|quotient map]] between algebraic objects (such as [[group (mathematics)|groups]]), this connection is called the [[lattice theorem]]: subgroups of {{mvar|G}} connect to subgroups of {{math|''G''/''N''}}, and the closure operator on subgroups of {{mvar|G}} is given by {{math|{{overline|''H''}} {{=}} ''HN''}}.