Galois connection

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Galois connection

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• Galois connexion

From Galois (attributive form of Galois theory) + connection; ultimately after French mathematician Évariste Galois. Coined by Norwegian mathematician Øystein Ore in 1944, Galois connexions, Transactions of the American Mathematical Society, 55, pages 493-513.

Galois connection (plural Galois connections)

1. (category theory, order theory) A type of correspondence between partially ordered sets (posets), also applicable to preordered sets.
   • 1986, Horst Herrlich, Miroslav Hušek, Galois Connections, Austin Melton, Mathematical Foundation of Programming Semantics: International Conference, Proceedings, Springer, Lecture Notes in Computer Science: 239, page 122,

     Define maps ${\displaystyle G:A\rightarrow B}$ and ${\displaystyle F:B\rightarrow A}$ by ${\displaystyle G(a)=\left\{y\!\in \!Y|\forall x\!\in \!a\ x\rho y\right\}}$ and ${\displaystyle F(b)=\left\{x\!\in \!X|\forall y\!\in \!b\ x\rho y\right\}}$. Then ${\displaystyle (F,G)}$ is called a Galois connection of the first kind.

   • 2001, J. Michael Dunn, Gary M. Hardegree, Algebraic Methods in Philosophical Logic, Oxford University Press, page 398:

     Finally, we discuss Galois connections. It is interesting to note that these definitions can be found in Birkhoff (1940, 1948, 1967) under the heading of "polarities," and Everett (1944) showed that all Galois connections defined on power sets can be obtained from polarities.

   • 2006, Radim Bělohlávek, Taťána Funioková, Vilém Vychodil, “Galois connections with Truth Stressers: Foundations for Formal Concept Analysis of Object-Attribute Data with Fuzzy Attributes”, in Bernd Reusch, editor, Computational Intelligence, Theory and Applications: International Conference, Proceedings, Springer,, page 205:

     Galois connections appear in several areas of mathematics and computer science, and their applications. A Galois connection between sets ${\displaystyle X}$ and ${\displaystyle Y}$ is a pair ${\displaystyle \left\langle \uparrow ,\downarrow \right\rangle }$ of mappings ${\displaystyle \uparrow }$ assigning subcollections of ${\displaystyle Y}$ to subcollections of ${\displaystyle X}$, and ${\displaystyle \downarrow }$ assigning subcollections of ${\displaystyle X}$ to subcollections of ${\displaystyle Y}$.

• adjunction

• isotone Galois connection, monotone Galois connection
• antitone Galois connection