A Category of Discrete Closure Systems

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 suggest the 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 ClosureSys of closure systems is shown to be cartesian closed. 1 Closure Systems By a discrete system we mean a set of elements, points, or other phenomena which we will generically call our universe, denoted by U. Individual points of U will be denoted by lower case letters: a, b, ..., p, q, ... ∈ U. By 2, we mean the powerset on U, or collection of all subsets of U. Elements of 2 we will denote by upper case letters: S, T, X, Y, Z. A closure system, C, is any collection of subsets X, Y, . . . Z ⊆ U, including U itself, which is closed under intersection. Subsets in C are said to be closed. If U = {a, b, c, d, e} then the collection of closed sets C1 = {Ø, {a}, {b}, {ab}, {bd}, {abc}, {abd}, {abce}, {abcde}, {abcdef}} (1) is a closure system. We require the empty set Ø to be included in all closure systems; although this will not be necessary until we define direct products in the category. A closure system can equivalently be defined as (U, φ), where φ is a closure operator satisfying four axioms. For all Y, Z ⊆ U, C0: Ø.φ = Ø C1: Y ⊆ Y.φ, C2: Y ⊆ Z implies Y.φ ⊂ Z.φ, and C3: Y.φ.φ = Y.φ. By C1, U itself must be closed. Here we are using a suffix operator notation, as we will throughout this paper. Read Y.φ as “Y closure”. A set Y is closed if Y = Y.φ. It is not hard to show that these two definitions of closure are equivalent. A closure operator/system can satisfy other axioms depending on the mathematical discipline. A topological closure is closed under union, or C4: (Y ∪ Z).φ = Y.φ ∪ Z.φ. The closure operator of linear systems, often called the spanning operator, satisfies the SteinitzMacLane exchange axiom