On fuzzy multi-value functions. I: Introduction and general properties (Q789307)

The basic concepts of fuzzy multi-valued function theory are introduced and studied in the context of applying fuzzy modelling to economical systems. The concepts of fuzzy multi-valued function (FMVF) (direct and converse) and a composite of two FMVFs and their properties are considered and examined (such as 1- and 2-monotonous property). The extension of a 'multi-valued fuzzy relation' is introduced. The definitions of fuzzy cone, convexity of a fuzzy cone, and of concave, superadditive and conical FMVF are given. The corresponding theorems are proved. The graph of a FMVF is determined by its membership function combined on the base of membership functions of the direct and converse FMVF. The graph of the composite of two FMVF's is defined, too. Their properties are presented without any assumption about the topological structure of the reference spaces considered.