An extension principle for closure operators
It is well known that a fuzzy set in an arbitrary universe can be decomposed into its weak level sets. Moreover the fuzzy set-theoretic operations can be expressed in terms of the weak level sets and the use of these level sets for the definitions of fuzzifications of crisp notions such as boundedness and convexity is also well-known. This paper clearly belongs to the third stage in the fuzzification process of classical mathematics namely the uniformization of the different fuzzification processes. The concept of a canonical extension is introduced as follows. Let \(A\) be a class of (crisp) subsets of a universe \(X\). Then the canonical extension \(A^*\) of \(A\) consists of the class of those fuzzy sets in \(X\) for which all their weak level sets belong to \(A\). It is proved that \(A^*\) is a fuzzy closure system iff \(A\) is a closure system. The authors illustrate the concept of a canonical extension by means of several examples such as Conrad's natural fuzzy topology, Pawlak's rough set theory and Zadeh's convex fuzzy sets in \(\mathbb{R}^n\). Finally the authors prove that the canonical extension of a classical closure system \(A\) coincides with the fuzzy closure system associated with \(A\).
scientific article; zbMATH DE number 902483