Rectangles-based discrete universal fuzzy integrals

Using hypergraphs of survival functions, we propose a rather general method for the construction of discrete fuzzy integrals. Our method is based on various rectangle decompositions of hypergraphs and on rectangle mappings suitably evaluating the rectangles of the considered decompositions. By means of appropriate binary aggregation functions we define two types of rectangle mappings and four types of discrete fuzzy integral constructions, and we also investigate the properties of the introduced integrals and the relationships between them. All the introduced methods based on non-overlapping rectangles coincide in the case of the product aggregation function, and then the related integral is the Choquet integral. Several examples are given.