Geometrical Interpretations of Interval-Valued Intuitionistic Fuzzy Sets: Reconsiderations and New Results

Intuitionistic fuzzy sets (IFSs), proposed in 1983, are one of the most viable and widely explored extensions of Zadeh’s fuzzy sets. In the decade following their introduction, they were extended to interval-valued IFSs (IVIFSs), temporal IFSs, IFSs of the second type (incorrectly called “Pythagorean fuzzy sets” by some authors) IFSs of n-th type, and IFSs over different universes. For each of these extensions, at least one geometrical interpretation has been defined, and for IVIFSs, at least seven different interpretations are known. In the present paper, revisiting some existing results on IVIFSs, some necessary modifications, additions, and corrections to the planar and spatial geometrical interpretations are introduced here for the first time. A new, eighth, geometrical interpretation of IVIFSs is proposed. A basic logic operation and two modal operators are illustrated and a comparison is made between the planar and the new “two-rods” geometrical interpretations of identical IVIFS elements. Finally, a new operator over IVIFSs is proposed for the first time, some of its properties are proven, and its geometrical interpretations are described.