It is well-known that Galois connections are useful in describing some situations that occur naturally in computer science and mathematics; and recently it has been shown that Lagois connections, which are closely related to Galois connections, are similarly useful. Thus, it is natural to ask if there are not common generalizations of Galois and Lagois connections which would be useful in both disciplines. In this paper we investigate several such generalizations. The primary one, called “connections”, was defined and first investigated in 1982 by H. Crapo. We present a hierarchy of connections from (general) connections to Lagois and Galois connections, and we establish properties of them. We also give examples in both computer science and mathematics.
key wordsconnection Galois connection Lagois connection semi-inverse poset system closure operator interior operator retraction operator
AMS subject classificationPrimary: 06A15, 06A10 Secondary: 68F05, 68F99, 54B99
Unable to display preview. Download preview PDF.
- J. Adámek, H. Herrlich and G.E. Strecker. Abstract and Concrete Categories. John Wiley & Sons, New York, 1990.Google Scholar
- T. S. Blyth and M. F. Janowitz. Residuation Theory. Pergamon Press, Oxford, 1972.Google Scholar
- G. Gierz et al. A Compendium of Continuous Lattices. Springer-Verlag, Berlin, 1980.Google Scholar
- H. Herrlich and M. Husek. Galois connections. Lecture Notes in Computer Science, No. 239, Springer-Verlag, Berlin, 1986, pp. 122–134.Google Scholar
- C.A.R. Hoare. The Mathematics of Programming. Clarendon Press, Oxford, 1986.Google Scholar
- H. Crapo. Ordered sets: retracts and connections. J. of Pure and Applied Algebra, 23 (1982) pp. 13–28.Google Scholar
- A. Melton, D. A. Schmidt, and G. E. Strecker. Galois connections and computer science applications. Lecture Notes in Computer Science, No. 240, Springer-Verlag, Berlin, 1986, pp. 299–312.Google Scholar
- A. Melton, B.S.W. Schröder, and G. E. Strecker. Lagois connections. (to appear).Google Scholar
- O. Ore. Galois connexions. Transactions of the American Mathematical Society, 55 (1944), pp. 493–513.Google Scholar
- G. Pickert. Bemerkungen über Galois-Verbindungen. Archiv der Mathematik, 3 (1952), pp. 285–289.Google Scholar
- J. Schmidt. Beiträge zur Filtertheorie, II. Mathematische Nachrichten, 10 (1953), pp. 197–232.Google Scholar
- Peter Wegner. Programming language semantics. Courant Computer Science Symposium 2, Prentice-Hall, Englewood Cliffs, New Jersey, 1972, pp. 149–248.Google Scholar