E. Tsiporkovahristoskova et al., A FUZZY INCLUSION BASED APPROACH TO UPPER INVERSE IMAGES UNDER FUZZY MULTIVALUED MAPPINGS, Fuzzy sets and systems, 85(1), 1997, pp. 93-108
Citations number
25
Categorie Soggetti
Computer Sciences, Special Topics","System Science",Mathematics,"Statistic & Probability",Mathematics,"Computer Science Theory & Methods
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.