Further Study of Multigranulation T-Fuzzy Rough Sets

The optimistic multigranulation T-fuzzy rough set model was established based on multiple granulations under T-fuzzy approximation space by Xu et al., 2012. From the reference, a natural idea is to consider pessimistic multigranulation model in T-fuzzy approximation space. So, in this paper, the main objective is to make further studies according to Xu et al., 2012. The optimistic multigranulation T-fuzzy rough set model is improved deeply by investigating some further properties. And a complete multigranulation T-fuzzy rough set model is constituted by addressing the pessimistic multigranulation T-fuzzy rough set. The full important properties of multigranulation T-fuzzy lower and upper approximation operators are also presented. Moreover, relationships between multigranulation and classical T-fuzzy rough sets have been studied carefully. From the relationships, we can find that the T-fuzzy rough set model is a special instance of the two new types of models. In order to interpret and illustrate optimistic and pessimistic multigranulation T-fuzzy rough set models, a case is considered, which is helpful for applying these theories to practical issues.