Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems

Images of fuzzy sets under fuzzy relations have been investigated mainly in two contexts: on the one hand, mostly under the term “full image” (Gottwald, 1993), they can be regarded as very general tools for fuzzy inference, leading to the so-called “compositional rule of inference” (Gottwald, 1993; Bauer et al., 1995). The theory of fuzzy relational equations makes direct use of this fundamental principle, too (Sanchez, 1984; Miyakoshi and Shimbo, 1985; di Nola et al., 1991; Gottwald, 1993; De Baets, 2000). On the other hand, under the term “extensional hull”, the image of a fuzzy set under a fuzzy equivalence relation yields the smallest fuzzy superset which is “closed” under the relation. This closedness property is usually called “extensionality” (Kruse et al., 1994). The concepts of extensionality and extensional hulls have turned out to be extremely useful, in particular when the analysis and interpretation of fuzzy partitions and controllers is concerned (Klawonn, 1993; Klawonn and Kruse, 1993; Klawonn and Castro, 1995; Klawonn et al., 1995).