Belief HyperFunctions, Belief SuperHyperFunctions, and DSm-Numbers

Abstract

Mathematical structures can be extended to hyperstructures and n-superhyperstructures via the powerset construction and its n-fold iteration. These generalized frameworks are well suited for modeling hierarchical relationships across diverse domains. Meanwhile, the notions of Hyperfunction and n-Superhyperfunction extend classical functions by mapping elements into iterated power sets, thereby capturing multi-level functional structures [1], [2]. Dempster–Shafer theory provides a robust approach to combine evidence from multiple sources, yielding belief and plausibility measures to model real-world uncertainty [3], [4]. Belief functions aggregate basic belief assignments over subsets to produce a rigorous lower bound on hypothesis probability given available evidence. In this paper, we develop the HyperFunction and SuperHyperFunction analogues of belief functions and related constructs, bridging evidence theory with hyperfunctional frameworks. Furthermore, a new numerical concept, called the DSm-Number, is introduced and formally defined.