A FUZZY INCLUSION BASED APPROACH TO UPPER INVERSE IMAGES UNDER FUZZY MULTIVALUED MAPPINGS

Fuzzy multivalued mappings have been introduced as a natural extension of classical multivalued mappings. It is well known that for single-v alued mappings we can consider direct and inverse images under such ma ppings. For multivalued mappings, however, Berge has introduced two ty pes of inverse images: lower and upper inverse images. In this paper, we concentrate on the upper inverse image, and present two new alterna tive definitions. These definitions have been developed using the conc ept of a fuzzy inclusion, more specifically a fuzzy inclusion defined from an implication operator. For the classical upper inverse image, i t is necessary to check inclusion of sets. It is therefore natural to use a fuzzy inclusion in the upper inverse image under fuzzy multivalu ed mappings. The new definitions are to some extent generalizations of the definition of Ottoy and Kerre, since a fuzzy inclusion can also b e recognized in their definition. The main difference is to be found i n the boundary behaviour, i.e. in the way in which cases where (non-)e mptiness plays a role are dealt with. The new definitions are exposed to a profound study. In order to simplify the study of their propertie s, a large number of new interesting properties of implication operato rs and fuzzy inclusions are presented first. The study of the upper in verse images includes monotonicity, complementation, interaction with union and intersection, specific behaviour on fuzzy constants, decompo sition laws and all kinds of relationships between images of cuts and cuts of images, and the like.