2005/01/01 by John L. Pfaltz, Pfaltz, John L.
Computer Science · #Advanced Algebra and Logic #Category #Rough Sets and Fuzzy Logic #antimatroid #closure #function
paper · doi:10.4230/dagsemproc.04351.4
openalex publication_date 2005/01/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Discrete systems such as sets, monoids, groups are familiar categories. The internal strucutre of the latter two is defined by an algebraic operator. In this paper we describe the internal structure of the base set by a closure operator. We illustrate the role of such closure in convex geometries and partially ordered sets and thus suggestthe wide applicability of closure systems. Next we develop the ideas of closed and complete functions over closure spaces. These can be used to establish criteria for asserting that "the closure of a functional image under f is equal to the functional image of the closure". Functions with these properties can be treated as categorical morphisms. Finally, the category "CSystem" of closure systems is shown to be cartesian closed.