Editing Galois connection (section)

====Connections on power sets arising from binary relations====
Suppose {{mvar|X}} and {{mvar|Y}} are arbitrary sets and a [[binary relation]] {{mvar|R}} over {{mvar|X}} and {{mvar|Y}} is given. For any subset {{mvar|M}} of {{mvar|X}}, we define {{math|''F''(''M''&hairsp;) {{=}} {&hairsp;''y'' ∈ ''Y'' {{!}} ''mRy'' ∀''m'' ∈ ''M''&hairsp;}.}} Similarly, for any subset {{mvar|N}} of {{mvar|Y}}, define {{math|''G''(''N''&hairsp;) {{=}} {&hairsp;''x'' ∈ ''X'' {{!}} ''xRn'' ∀''n'' ∈ ''N''&hairsp;}.}} Then {{mvar|F}} and {{mvar|G}} yield an antitone Galois connection between the power sets of {{mvar|X}} and {{mvar|Y}}, both ordered by inclusion ⊆.{{sfn|Birkhoff|1940|loc=§32; 3rd edition (1967): Ch. V, §7 and §8}}

Up to isomorphism ''all'' antitone Galois connections between power sets arise in this way. This follows from the "Basic Theorem on Concept Lattices".<ref>Ganter, B. and Wille, R. ''Formal Concept Analysis -- Mathematical Foundations'', Springer (1999), {{ISBN|978-3-540-627715}}</ref> Theory and applications of Galois connections arising from binary relations are studied in [[formal concept analysis]]. That field uses Galois connections for mathematical data analysis.  Many algorithms for Galois connections can be found in the respective literature, e.g., in.<ref>Ganter, B. and Obiedkov, S. ''Conceptual Exploration'', Springer (2016), {{ISBN|978-3-662-49290-1}}</ref>

The [[General Concept Lattice|general concept lattice]] in its primitive version incorporates both the monotone and antitone Galois connections to furnish its upper and lower bounds of nodes for the concept lattice, respectively.<ref name=":1">{{Cite journal |last1=Liaw |first1=Tsong-Ming |last2=Lin |first2=Simon C. |date=2020-10-12 |title=A general theory of concept lattice with tractable implication exploration |url=https://www.sciencedirect.com/science/article/pii/S0304397520302826 |url-status=live |journal=Theoretical Computer Science |language=en |volume=837 |pages=84–114 |doi=10.1016/j.tcs.2020.05.014 |issn=0304-3975 |s2cid=219514253 |archive-url=https://web.archive.org/web/20200528022615/https://www.sciencedirect.com/science/article/pii/S0304397520302826 |archive-date=2020-05-28 |access-date=2023-07-19}}</ref>