Meaning of Galois connection | Babel Free

Definitions

A type of correspondence between partially ordered sets (posets), also applicable to preordered sets.

Examples

“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 G:A→B and F:B→A by G(a)=y!∈!Y|∀x!∈!axρy and F(b)=x!∈!X|∀y!∈!bxρy. Then (F,G) is called a Galois connection of the first kind.”

“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.”

“Galois connections appear⟳ in several areas of mathematics and computer science, and their applications. A Galois connection between sets X and Y is a pair #92;left#92;langle#92;uparrow,#92;downarrow#92;right#92;rangle of mappings #92;uparrow assigning subcollections of Y to subcollections of X, and #92;downarrow assigning subcollections of X to subcollections of Y.”

CEFR level