New category of the Fuzzy d-algebras

Abstract

The aim of this work is to introduce a new category of Fuzzy d-algebra called Γ-Fuzzy d-algebra. This notion is more useful than Fuzzy d-algebra because in the old case the degree of membership of any $x \in X$ in the Fuzzy set does not depend on the structure of d-algebra X. Therefore, in this work, we introduced a new construction of Fuzzy d-algebra that is depended on structure of d-algebra. Also, we show that for any d-algebra there is a Γ-Fuzzy d-algebra. Further, this new construction helps us to find the interesting relation between Γ-Fuzzy d-algebra and edge d-algebra in this work. For any d-algebra, we determined unique Γ-Fuzzy topological space $\lpar X\comma \; \, \Gamma ^f\rpar $ using Γ-Fuzzy d-algebra. For any $f \in X$, a new class of d-algebra $\lpar \Psi \lpar X\rpar \comma \; \varpi \comma \; \lpar X\comma \; \otimes _f\comma \; \, \, f\rpar ^{\mu ^{ \otimes _f}}\rpar $ is introduced, where $\Psi \lpar X\rpar $ is the set of all equivalence classes $\lpar X\comma \; \ast \comma \; \, \, f\rpar ^\lambda $. Finally, a new class of intuitionistic fuzzy set is called intuitionistic Γ-Fuzzy d-algebra is investigated and discussed.

Keywords

• d-Algebra
• Fuzzy sets theory
• edge d-algebra
• absolutely Γ-Fuzzy topological space