Lukasiewicz intuitionistic fuzzy sets and its application in BCK-algebras and BCI-algebras

Abstract

Using the concepts of \L{}ukasiewicz $t$-norm and \L{}ukasiewicz $t$-conorm, the \L{}ukasiewicz intuitionistic fuzzy set based on an intuitionistic fuzzy set is introduced and applied it to BCK-algebras and BCI-algebras. The notion of (strong) \L{}ukasiewicz intuitionistic fuzzy subalgebra is introduced and its various properties are investigated. Characterizations of \L{}ukasiewicz intuitionistic fuzzy subalgebras are discussed. Conditions for \L{}ukasiewicz intuitionistic fuzzy set to be a \L{}ukasiewicz intuitionistic fuzzy subalgebra are explored, and the conditions under which \L{}ukasiewicz intuitionistic fuzzy subalgebra becomes strong are provided. Three types of subsets so called \L{}ukasiewicz $\in$-set, \L{}ukasiewicz $q$-set and \L{}ukasiewicz $O$-set are established, and the conditions under which they can be subalgebras are displayed.