Galois connection

In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets ("posets"). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. They find applications in various mathematical theories as well as in the theory of programming.

References

A freely available introduction to Galois connections, presenting many examples and results. Also includes notes on the different notations and definitions that arose in this area:

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• M. Erné, J. Koslowski, A. Melton, G. E. Strecker, A primer on Galois connections, in: Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work, Annals of the New York Academy of Sciences, Vol. 704, 1993, pp. 103-125. Available online in various file formats: PS.GZ PS

The following standard reference books also include Galois connections using modern notation and definitions:

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• B. A. Davey and H. A. Priestley: Introduction to lattices and Order, Cambridge University Press, 2002.
• G. Gierz, K. H. Hoffmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: Continuous Lattices and Domains, Cambridge University Press, 2003.

Finally, some publications using the original (antitone) definition:

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• Garrett Birkhoff: Lattice Theory, Amer. Math. Soc. Coll. Pub., Vol 25, 1940
• Oystein Ore: Galois Connexions, Transactions of the American Mathematical Society 55 (1944), pp. 493-513