A multi-granulation T-rough set model and its properties

This paper works in the setting of multi-granulation rough sets on two universes. We combine multi-granulation ideas with set-valued mappings and introduce a multi-granulation T-rough set model built on set-valued mappings. Four inverse approximation operators are defined: a pessimistic upper inverse operator, a pessimistic lower inverse operator, an optimistic upper inverse operator, and an optimistic lower inverse operator. Using these operators, we set up a two-universe framework for multi-granulation T-rough sets. We then spell out the basic properties of the operators, prove several theorems, and clarify how the different operators relate to each other. A few examples are included to show how the model works in practice. The model extends rough set theory on two universes and gives a new way to describe information that comes from multiple sources and multiple granular levels.