Basic Properties Of Fuzzy Approximation Operators Under Consistent Functions

With the development of science and technology, uncertainty problems have become a new demand in daily life. How to effectively deal with uncertainty problems has become a new field of research. In many theories, Both fuzzy sets and rough sets are mathematical tool to deal with uncertainty, but they are different from each other. The theory of fuzzy sets is proposed by L. A. Zadeh in 1965 and the theory of rough sets is proposed by Pawlak in 1982, the combination of rough sets and fuzzy sets lead to the notions of fuzzy rough sets and rough fuzzy sets. The two theories have been widely studied and many important results are achieved. The topics on approximation operators have been one of the most important directions and obtained much attention, and approximation operators are more and more widely applied in the field of artificial intelligence such as machine learning, information exchange and relational database. In this paper, we use the concepts of consistent functions as a tool to study some basic properties of approximation operators. First, we propose the concept of consistent functions and study their basic properties, then some properties of fuzzy approximation operators under the consistent functions are studied. Furthermore, we introduce consistent functions into fuzzy decision information systems and study basic properties of approximation operators. Finally, we study the attribute reduction in fuzzy decision information systems. The main structure of this paper is organized as follows.1. We review the concepts of the expansion principle and relational mappings, and introduce the concepts of fuzzy approximation operators and consistent functions related to fuzzy relations.2. We study basic properties of consistent functions and discuss some properties of fuzzy approximation operators under consistent functions.3. We introduce consistent functions into fuzzy decision information systems and study basic properties of approximation operators in fuzzy decision information systems. Furthermore, we study the attribute reduction in fuzzy decision information systems.