A primer on Galois connections. (English) Zbl 0809.06006

Erné, M.; Koslowski, J.; Melton, A.; Strecker, George E.

Andima, Susan (ed.) et al., Papers on general topology and applications. Proceedings of the seventh summer conference in honor of Mary Ellen Rudin and her work, held at the University of Wisconsin, Madison, WI, USA, June 26 to 29, 1991. New York, NY: The New York Academy of Sciences,. Ann. N. Y. Acad. Sci. 704, 103-125 (1993).

Summary: The rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) are provided, together with many examples and applications. Galois connections occur in profusion and are well known to most mathematicians who deals with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear, and thus, the whole situation typically becomes much easier to understand.
For the entire collection see [Zbl 0801.00034].

MSC:

06A15	Galois correspondences, closure operators (in relation to ordered sets)
54-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology
06-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
06A06	Partial orders, general
54B99	Basic constructions in general topology
54H99	Connections of general topology with other structures, applications

Keywords:

closure operation; interior operation; polarity; axiality; residuated semigroups; Galois connections; residuation theory