Independence of axiom sets on intuitionistic fuzzy rough approximation operators

Algebraic approach is widely used in the study of rough sets. In the algebraic approach, one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators. Axiomatic approaches have been used in the study of intuitionistic fuzzy rough sets. Intuitionistic fuzzy rough approximation operators are defined by axioms, and different axiom sets of lower and upper intuitionistic fuzzy rough approximation operators guarantee the existence of different types of intuitionistic fuzzy relations. However, the independence of the axioms in each of those axiom sets has not been proved. If the axioms are not independent, one of the axioms is redundant. In this paper, we prove that no axioms can be derived from other axioms, which means the axioms in the axiom set are independent. For each axiom set, we examine the independence of axioms and present the independent axiom sets characterizing intuitionistic fuzzy rough approximation operators. Thus, we improve the intuitionistic fuzzy rough set theory.