Generating embedded type-1 fuzzy sets by means of convex combination

This paper presents a summarized characterization of embedded type-1 fuzzy sets (ET1FS) by using the classical concept of convex combination given by Zadeh. ET1FSs are important when defining the centroid of an interval-valued fuzzy set (IVFS) and some type-reduction methods proposed in the literature. We will show that any ET1FS of an IVFS, with no assumption whatsoever about the universal set (either continuous or discrete), can be expressed always as the convex combination of three type-1 fuzzy sets (T1FS) whose membership functions (MF) correspond to: the lower and upper MFs (of the IVFS), and a third arbitrary MF.