L-fuzzifying approximation operators in fuzzy rough sets

In rough set theory, lower and upper approximation operators are two primitive notions. Various fuzzy generalizations of lower and upper approximation operators have been introduced over the years. Considering L being a completely distributive De Morgan algebra, this paper mainly proposes a general framework of L-fuzzifying approximation operators in which constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper L-fuzzifying approximation operators is defined. The connections between L-fuzzy relations and L-fuzzifying approximation operators are examined. In the axiomatic approach, various types of L-fuzzifying rough sets are proposed and L-fuzzifying approximation operators corresponding to each type of L-fuzzy relations as well as their compositions are characterized by single axioms. Moreover, the relationships between L-fuzzifying rough sets and L-fuzzifying topological spaces are investigated. It is shown that there is a one-to-one correspondence between reflexive and transitive L-fuzzifying approximation spaces and saturated L-fuzzifying topological spaces.