On the structure of the classes of stable and pure fuzzy sets. (English) Zbl 0930.03077
Summary: Two interesting classes of sets with respect to a multivalued mapping are considered: the class of stable sets and the class of pure sets. In addition to the results existing in the literature, several new properties of stable and pure sets are established. A generalization of the concepts of stable and pure sets to fuzzy multivalued mappings is developed. The properties of stable and pure fuzzy sets are studied extensively. It is obtained that the stable (resp. pure) fuzzy sets with respect to a normalized fuzzy multivalued mapping constitute a complete lattice and also a stratified fuzzy topology. The notions of level stable and level pure fuzzy sets are introduced. Interactions between stability (resp. purity) and level stability (resp. level purity) are established and expressed in terms of relationships between particular stratified fuzzy topologies. Some additional relationships between the stability (resp. purity) of a fuzzy set with respect to a multivalued mapping and the stability (resp. purity) of its cuts with respect to this mapping are established. By imposing certain conditions on the triangular norm and the implication operator involved a one-to-one correspondence between the class of stable fuzzy sets and the class of pure fuzzy sets with respect to a normalized fuzzy multivalued mapping that has a normalized inverse is obtained. In the last section, it is shown that a normalized fuzzy multivalued mapping is continuous with respect to the stratified fuzzy topologies constituted by the stable fuzzy sets and the pure fuzzy sets with respect to this fuzzy multivalued mapping.
Keywords:
continuity; stable sets; pure sets; fuzzy multivalued mappings; complete lattice; stratified fuzzy topology; level stability; level purity; triangular normReferences:
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