Implication Operators Based on Rough Set Model over Boolean Algebras

Some algebraic properties of rough approximation operators over Boolean algebras are investigated and the concept of rough Stone algebra (by J.Pomykala and J.A.Pomykala) and rough Nelson algebra (it is also called rough sets system by P.Pagliani) are generalized. From two directions, the rough implication operators based on rough set model over Boolean algebras are studies as following: (1) from the description of the pairs (low approximation, upper approximation) of rough sets, the implication operator rarr_D is introduced by modifying the method by I.Duntsch; (2) from the description of the pairs (low approximation, the complement of upper approximation) of rough sets, a pair of implication operator (rarr_L, rarr_G) is introduced using the method by G.Cattaneo and D.Ciucci. Basis of this, the important results are proved: the rough Stone algebra over Boolean algebra with rarr_D form a IMTL-algebra; the rough Nelson algebra over Boolean algebra with (rarr_L, rarr_G) form a HW(Heyting Wajsberg) algebra. Finally, the relationship between rough logic and fuzzy logic is discussed and a new way for studying rough logic is pointed out.