The ( 2 , 3 ) -fuzzy set and its application in BCK-algebras and BCI-algebras

Abstract

It is well-known that an intuitionistic fuzzy set is a generalization of a fuzzy set. Intuitionistic fuzzy sets deal with two types of fuzzy sets, namely membership function and non-membership function, under the condition that the sum of the membership degree and non-membership degree is less than or equal to \(1\). If the sum of membership degree and non-membership degree is greater than or equal to \(1\), the intuitionistic fuzzy set feels limited in its role. The Pythagorean fuzzy set that emerged to overcome these limitations is the generalization of the intuitionistic fuzzy set. As another form of generalization of the intuitionistic fuzzy set, the concept of the \((2,3)\)-fuzzy set is introduced in this article and several attributes are investigated. The semigroup structure is assigned to the collection of \((2,3)\)-fuzzy sets. The concepts of \((2,3)\)-fuzzy subalgebra for BCK/BCI-algebra and closed \((2,3)\)-fuzzy subalgebra for BCI-algebra are introduced and their properties are investigated. The relationship between the \((2,3)\)-fuzzy subalgebra and the degree function is discussed. A new \((2,3)\)-fuzzy subalgebra is generated using the given \((2,3)\)-fuzzy subalgebra. The union and intersection of \((2,3)\)-fuzzy subalgebras are addressed, and the characterization of the \((2,3)\)-fuzzy subalgebra using the \((2,3)\)-cutty set is addressed. Conditions for closing the \((2,3)\)-fuzzy subalgebra in the BCI-algebra are retrieved.