Generalization of intuitionistic fuzzy submodules of a module by using triangular norms and conorms and (T,S)-L subrings

Abstract

This study consists of two parts. In first part; It is built on the definition of Intuitionistic Fuzzy Submodules of a Module. Many researchers have used the definition of Atanassov’s (Fuzzy Sets Syst 20:87–96, [1]) Intuitionistic fuzzy sets definition to move the definitions in classical algebra to intuitionistic fuzzy algebra. Davvaz et al. (Inf Sci 176:1447–1454, [2]) defined the Intuitionistic fuzzy submodules of a module. They used minimum and maximum operations to give that definition. In this study we replace minimum operation with triangular norms and maximum operation with triangular conorms for giving the definition of Intuitionistic (T, S)-fuzzy submodule of a module. By using this definition, we move some definition and theorems in classical algebra to Intuitionistic fuzzy algebra. In the second part It is built on the definition of intuitionistic L-fuzzy rings and ideals. Many researchers have used the definition of Atanassov’s (Fuzzy Sets Syst 20:87–96, [1]) intuitionistic fuzzy sets to move the definitions in classical algebra to intuitionistic fuzzy algebra (Davvaz et al. in Inf Sci 176:1447–1454, [2]; Çuvalcıoğlu et al. in Notes Intuitionistic Fuzzy Sets 20:9–16, [3]; Çuvalcıoğlu and Aykut in NIFS 20:57–61, [4]; Isaac and Pearly in Int J Math Sci Appl 1:1447–1454, [5]). When K. Atannassov gave the definition of intuitionistic fuzzy sets he used the closed interval [0, 1]. Then Meena and Thomas (Int Math Forum 6:2561–2572, [6]) replaced the closed interval [0, 1] with L-lattice. In that study they used ∧ ∧-infimum and ∨-supremum operations to give the intuitionistic L-fuzzy rings and intuitionistic L-fuzzy ideals. In this study we replace ∧-infimum with triangular norms and we replace ∨-supremum with triangular conorms and give the definition of intuitionistic (T,S)-L fuzzy rings and ideals. By using this definitions, we move some definition and theorems in classical algebra to intuitionistic fuzzy algebra.