Lukasiewicz anti fuzzy set and its application in BE-algebras

‎The idea of  Lukasiewicz $t$-conorm is used to construct the concept of  Lukasiewicz anti fuzzy sets based on a given anti fuzzy set‎, ‎and it is applied to BE-algebras‎. ‎The notion of  Lukasiewicz anti fuzzy BE-ideal is introduced‎, ‎and its properties are investigated‎. ‎The conditions under which  Lukasiewicz anti fuzzy set will be  Lukasiewicz anti fuzzy BE-ideal are explored‎, ‎and the relationship between anti fuzzy BE-ideal and  Lukasiewicz anti fuzzy BE-ideal are discussed‎. ‎Three types of subsets so called $\lessdot$-subset‎, ‎$\Upsilon$-subset‎, ‎and anti subset are constructed‎, ‎and the conditions under which they can be BE-ideals are explored‎.