ON THE STRUCTURE OF THE CLASSES OF STABLE AND PURE FUZZY-SETS

In this paper, two interesting classes of sets with respect to a multi valued mapping are considered: the class of stable sets and the class of pure sets. In addition to the results existing in the literature, s everal new properties of stable and pure sets are established. A gener alization of the concepts of stable and pure sets to fuzzy multivalued mappings is developed. The properties of stable and pure fuzzy sets a re studied extensively. It is obtained that the stable (resp. pure) fu zzy sets with respect to a normalized fuzzy multivalued mapping consti tute a complete lattice and also a stratified fuzzy topology. The noti ons of level stable and level pure fuzzy sets are introduced. Interact ions between stability (resp. purity) and level stability (resp. level purity) are established and expressed in terms of relationships betwe en particular stratified fuzzy topologies. Some additional relationshi ps between the stability (resp. purity) of a fuzzy set with respect to a multivalued mapping and the stability (resp. purity) of its cuts wi th respect to this mapping are established. By imposing certain condit ions on the triangular norm and the implication operator involved a on e-to-one correspondence between the class of stable fuzzy sets and the class of pure fuzzy sets with respect to a normalized fuzzy multivalu ed mapping that has a normalized inverse is obtained. In the last sect ion, it is shown that a normalized fuzzy multivalued mapping is contin uous with respect to the stratified fuzzy topologies constituted by th e stable fuzzy sets and the pure fuzzy sets with respect to this fuzzy multivalued mapping. (C) 1998 Elsevier Science B.V. All rights reserv ed.