Intuitionistic fuzzy normed rings

Abstract

The notion of intuitionistic fuzzy set was first introduced as a generalization of fuzzy set. Several researchers applied this notion in many branches of mathematics, primarily in algebra and defined the notions of intuitionistic fuzzy subgroups, intuitionistic fuzzy subrings and intuitionistic fuzzy ideals. In this thesis, we introduce the notion of intuitionistic fuzzy normed rings. We develop a new structure of fuzzy rings which will be called an intuitionistic fuzzy normed ring. The main purpose of the thesis is to introduce intuitionistic fuzzy normed subrings and other subrings related to intuitionistic fuzzy normed subring. Also, to define intuitionistic fuzzy normed ideals and generalize the notions of intuitionistic fuzzy normed prime and maximal ideals. The discussion in this thesis is divided into two parts. In the first part, we introduce the notion of intuitionistic fuzzy normed subrings. Next, we discuss some operations of intuitionistic fuzzy normed subrings such as subset, equality, complement, intersection and union. Further, several properties and illustrative examples are discussed. Moreover, we introduce and discuss the image and the inverse image of intuitionistic fuzzy normed subrings and study their elementary properties. Additionally, we introduce the notions of intuitionistic fuzzy normed normal subrings and intuitionistic anti fuzzy normed normal subrings. Besides, we characterize direct product of intuitionistic fuzzy normed normal subrings and direct product of finite intuitionistic fuzzy normed normal subrings. We generalize the relation between intuitionistic characteristic function and direct product of intuitionistic fuzzy normed normal subrings and examine some fundamental properties. Similarly, we characterize direct product of intuitionistic anti fuzzy normed normal subrings and direct product of finite intuitionistic anti fuzzy normed normal subrings. Moreover, we define intuitionistic anti characteristic function and establish the relation between intuitionistic anti characteristic function and direct product of intuitionistic anti fuzzy normed normal subrings and examine some fundamental operations. The inspiration of the second part of this thesis is to introduce the idea of intuitionistic fuzzy normed ideals. Thus, we begin the second part by introducing the concept of intuitionistic fuzzy ideals and basic properties of this concept are examined. Furthermore, we characterise the notions of intuitionistic fuzzy prime ideals and intuitionistic fuzzy maximal ideals and some related properties are investigated. Finally, we define intuitionistic anti fuzzy normed ideals and define the anti level subsets and identifies the relationship between these sets and intuitionistic anti fuzzy normed ideals. In addition, we define the anti image of intuitionistic anti fuzzy normed ideals and discuss the inverse image and anti image for intuitionistic anti fuzzy normed ideals under epimorphism mapping.