Some novel results on semiring hypergraph rough sets (Nonlinear Studies, 2025)
Classical rough set models usually tend to be restrictive in their application with overlapping and uncertain data, rendering them largely incompatible with higher order relationships involving especially complex relational structures. This paper introduces a new algebraic foundation of the hypergraph rough set (HRS) theoretic model based on commutative monoids and semirings to investigate multiple relationships involving a broad class of data structures. We showed that the disjoint union set operation $\veebar$ and disjoint intersection set operation $\barwedge$ over power sets forms a commutative monoid under the corresponding relation, whereas the explore intersection $\barwedge$ and disjoint union $\veebar$ operations together constitute a distributive semiring closed under addition of identities. The uni-modal semiring algebra can aid in interpretability, data structure classification and the computational capabilities for performing rough set operations. In addition, we discuss a case study of medical diagnosis to illustrate how the HRS model can process uncertain and overlapping patient information. We also took into account the scalability of the proposed approach in dynamic environments and real-time decision making. Finally, we implemented the semiring based framework of HRS as a formalized and systematic analysis model for rich relational structures and specific relationships under uncertain data systems.