ℒ-fuzzy Annihilators in Residuated Lattices

Abstract

ABSTRACT In this paper, we provide a new characterization of ℒ -fuzzy ideals of residuated lattices, which allows us to describe ℒ -fuzzy ideals generated by ℒ -fuzzy sets. Thanks to the latter, we endow the lattice of ℒ -fuzzy ideals of a residuated lattice with suitable operations. Moreover, we introduce the notion of ℒ -fuzzy annihilator of an ℒ -fuzzy subset of a residuated lattice with respect to an ℒ -fuzzy ideal and investigate some of its properties. To this extent, we show that the set of all ℒ -fuzzy ideals of a residuated lattice is a complete Heyting algebra. Furthermore, we define some types of ℒ -fuzzy ideals of residuated lattices, namely stable ℒ -fuzzy ideals relative to an ℒ -fuzzy set, and involutory ℒ -fuzzy ideals relative to an ℒ -fuzzy ideal. Finally, we prove that the set of all stable ℒ -fuzzy ideals relative to an ℒ -fuzzy set is also a complete Heyting algebra, and that the set of involutory ℒ -fuzzy ideals relative to an ℒ -fuzzy ideal is a complete Boolean algebra.