Fuzzy, Neutrosophic, Plithogenic, Rough, Granular, and Functorial Ordered and Ranking Numbers

This paper develops a unified framework for uncertainty numbers by introducing ordered and ranking structures across six paradigms: fuzzy, neutrosophic, plithogenic, rough, granular, and functorial numbers. We define ordered variants via monotone boundary curves, chains of approximations, and diagrammatic (functorial) morphisms, and we prove that ordered neutrosophic and plithogenic numbers strictly generalize ordered fuzzy numbers. On the comparative side, we specify ranking functionals that are monotone with respect to natural dominance relations and show that ranking neutrosophic numbers recover classical rankings of fuzzy numbers, while ranking plithogenic numbers subsume both. We also formalize ordered and ranking versions of rough and granular numbers and establish retention of their native structures under projection. Finally, we introduce Ordered/Ranking Functorial Numbers, which organize all models as semiring-valued diagrams, yielding embedding and stability theorems and illustrative examples.