A Primer on Galois Connections

ABSTRACT. 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 deal 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.

[1]  F. William Lawvere,et al.  Adjointness in Foundations , 1969 .

[2]  C. J. Everett.  Closure operators and Galois theory in lattices , 1944 .

[3]  Marcel Erné.  ORDER EXTENSIONS AS ADJOINT FUNCTORS , 1986 .

[4]  David A. Schmidt,et al.  Calois Connections and Computer Science Applications , 1985, CTCS.

[5]  M. Stone.  The theory of representations for Boolean algebras , 1936 .

[6]  Bernhard Banaschewski,et al.  Cauchy Points of Metric Locales , 1989, Canadian Journal of Mathematics.

[7]  G. Grätzer.  General Lattice Theory , 1978 .

[8]  Horst Herrlich,et al.  Galois connections categorically , 1990 .

[9]  Emil Popescu,et al.  On Galois Connexions , 1994 .

[10]  Karl H. Hofmann,et al.  The algebraic theory of compact Lawson semilattices : applications of Galois connections to compact semilattices , 1976 .

[11]  Irène Guessarian.  On Continuous Completions , 1979, Theoretical Computer Science.

[12]  William Graves,et al.  The Category of Complete Algebraic Lattices , 1972, J. Comb. Theory, Ser. A.

[13]  G. Strecker,et al.  A Factorization of the Pumpl Un-r Ohrl Connection , 1992 .

[14]  George N. Raney,et al.  A subdirect-union representation for completely distributive complete lattices , 1953 .

[15]  K. I. Rosenthal.  Quantales and their applications , 1990 .

[16]  James W. Thatcher,et al.  A Uniform Approach to Inductive Posets and Inductive Closure , 1977, MFCS.

[17]  Marcel Erné,et al.  The category of Z-continuous posets , 1983 .

[18]  Orrin Frink,et al.  Pseudo-complements in semi-lattices , 1962 .

[19]  M. Erné,et al.  Completions for partially ordered semigroups , 1986 .

[20]  Helmut Röhrl,et al.  Separated totally convex spaces , 1985 .

[21]  Marcel Erné,et al.  Algebraic Ordered Sets and Their Generalizations , 1993 .

[22]  George E. Strecker,et al.  A factorization of the Pumplün-Röhrl connection , 1992 .

[23]  Wojciech A. Trybulec.  Partially Ordered Sets , 1990 .

[24]  Dan Novak.  Generalization of continuous posets , 1982 .

[25]  H. Sonner,et al.  Die Polarität zwischen topologischen Räumen und Limesräumen , 1953 .

[26]  Zahava Shmuely,et al.  The structure of Galois connections. , 1974 .

[27]  Günter Pickert.  Bemerkungen überGalois-Verbindungen , 1952 .

[28]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[29]  R. Wiegandt,et al.  Connectednesses and disconnectednesses in topology , 1975 .

[30]  Jürgen Schmidt,et al.  Beiträge zur Filtertheorie. II , 1952 .

[31]  M. Stone,et al.  The Theory of Representation for Boolean Algebras , 1936 .

[32]  Glynn Winskel,et al.  Relating Two Models of Hardware , 1987, Category Theory and Computer Science.

[33]  R. Sikorski,et al.  The mathematics of metamathematics , 1963 .

[34]  Gerhard Preuß,et al.  Eine Galois-Korrespondenz in der Topologie , 1971 .

[35]  John R. Isbell,et al.  Atomless Parts of Spaces. , 1972 .

[36]  G. Strecker,et al.  Closure Operators and Polarities a , 1993 .