On the simplicial structure of uncertain information

Abstract

The mathematical representation of uncertainty has led to a proliferation of preference structures, such as interval-valued fuzzy sets, intuitionistic fuzzy sets, and various granular models. While these extensions are often studied independently, they share profound geometric and topological foundations. This paper provides a unifying framework by identifying these disparate structures with the simplicial geometry of $n$-dimensional fuzzy sets. We first conduct an extensive revision of both classical and modern preference structures, demonstrating that they are distinct semantic interpretations of the same underlying topological objects within the lattice $L_n$. Building on this unification, we introduce a new, highly interpretable preference structure based on Deck-of-Cards membership functions. This approach generalizes the revised models by providing a flexible mechanism to represent complex membership degrees through monotonic sequences. Furthermore, we establish a formal simplicial structure for the set of multidimensional fuzzy sets $L_\infty$. By employing face and degeneracy maps, we demonstrate how this framework unifies existing models into a single simplicial set, allowing for the consistent transformation of information across different levels of granularity. The examples provided illustrate the utility of this simplicial connection in several contexts, offering a robust topological foundation for future developments in fuzzy set theory.