Extension principles for interval-valued intuitionistic fuzzy sets and algebraic operations

The Atanassov's intuitionistic fuzzy (IF) set theory has become a popular topic of investigation in the fuzzy set community. However, there is less investigation on the representation of level sets and extension principles for interval-valued intuitionistic fuzzy (IVIF) sets as well as algebraic operations. In this paper, firstly the representation theorem of IVIF sets is proposed by using the concept of level sets. Then, the extension principles of IVIF sets are developed based on the representation theorem. Finally, the addition, subtraction, multiplication and division operations over IVIF sets are defined based on the extension principle. The representation theorem and extension principles as well as algebraic operations form an important part of Atanassov's IF set theory.