Extensionality in Graded Properties of Fuzzy Relations

New definitions of graded reflexivity, symmetry, transitivity, antisymmetry, and functionality of fuzzy relations are proposed which are relative to an indistinguishability relation E on the universe of discourse. It is shown that if considered nongraded (i.e., either fully present or else fully absent), the new definitions reduce to the usual ones under full extensionality of the relation w.r.t. E. However, if graded properties of R (e.g., transitivity to some degree) are taken into account, the new definitions have to be distinguished from the conjunction of the original property and E-extensionality of R. Some arguments and results are given which suggest that the new concepts are well-motivated.