Abstract: The existing research conclusions on the principle of rough fuzzy set extension are based on the representation of fuzzy subset union. In this paper, we introduce the decomposition theorems for the representation of upper cut set, lower cut set, lower heavy cut set and upper heavy cut set, and the corresponding four cut sets based on the representation of intersection, give the extension principle of the rough fuzzy set based on the four cut sets with theoretical proof in detail.

Key words: rough fuzzy set, cut set, representation of intersection, decomposition theorem, extension principle