On intuitionistic fuzzy rough set algebras

An intuitionistic fuzzy rough set is a pair of intuitionistic fuzzy sets resulting from the approximation of an intuitionistic fuzzy set in an intuitionistic fuzzy approximation space. An intuitionistic fuzzy rough set algebra is an intuitionistic fuzzy set algebra with added dual pair of intuitionistic fuzzy rough approximation operators. In this paper, we study the mathematical structures of intuitionistic fuzzy rough set algebras. We first define the concept of intuitionistic fuzzy rough set algebras by the axiomatic approach. We then examine the properties of intuitionistic fuzzy rough approximation operators in different types of intuitionistic fuzzy rough set algebras. We also prove that if a system (LF(U), ∩, ∪, ∼, L, H) is an (respectively, a serial, a reflexive, a symmetric, a transitive, a topological, a similarity) intuitionistic fuzzy rough set algebra, then the derived system (LF(U), ∩, ∪, ∼, LL, HH) is also an (respectively, a serial, a reflexive, a symmetric, a transitive, a topological, a similarity) intuitionistic fuzzy rough set algebra.