A note on direct product of complex intuitionistic fuzzy subfield

Abstract

This paper presents the concepts of a complex intuitionistic fuzzy subfield (CIFSF) and the direct product of a complex intuitionistic fuzzy subfield which is generalized from the concept of a complex fuzzy subfield by adding the notion of intuitionistic fuzzy into a complex fuzzy subfield. The main contribution and originality of this research are adding the non-membership term to the definition of a complex fuzzy subfield that assigns for any element a complex-valued grade. We expand the complex fuzzy subfield and obtain a new structure called CIFSF. This new concept is innovative in that it may attain a wider range of values for both membership and non-membership functions where these functions are expanded to the unit disc in the complex plane. Furthermore, we discuss that the direct product of two CIFSFs is CIFSF, and some related properties are investigated. In addition, we present the definition of necessity and possibility operators on the direct product of CIFSF, and some associated theorems are given. Finally, we propose the level subsets of the direct product of two complex intuitionistic fuzzy subsets of a field and prove that the level subset of the direct product of two CIFSFs is a subfield and discuss some related results.