Decomposition of Bipolar Pythagorean Fuzzy Matrices

This paper presents novel findings on modal operators through the use of max-min composition, analyzing properties such as reflexivity, symmetry, transitivity, and idempotency related to necessity and possibility. It explores the necessary and sufficient conditions for transitive and \(c\)-transitive closure matrices using modal operators. Additionally, a new composition operator, labeled as `\(\wedge_{m}\)' is introduced and its algebraic properties are thoroughly discussed. The study also achieves a decomposition of a BPyFM utilizing the new composition operator and modal operators.