Lattices of (Generalized) Fuzzy Ideals in Double Boolean Algebras

Abstract

This paper develops the notion of fuzzy ideal and generalized fuzzy ideal on double Boolean algebra (dBa)‎. ‎According to Rudolf Wille‎, ‎a double Boolean algebra $\underline{D}:=(D‎, ‎\sqcap‎, ‎\sqcup‎, ‎\neg‎, ‎\lrcorner‎, ‎\bot‎, ‎\top)$ is an algebra of type $(2‎, ‎2‎, ‎1‎, ‎1‎, ‎0‎, ‎0),$ which satisfies a set of properties‎. ‎This algebraic structure aimed to capture the equational theory of the algebra of protoconcepts‎. ‎We show that collections of fuzzy ideals and generalized fuzzy ideals are endowed with lattice structures‎. ‎We further prove that (by isomorphism) lattice structures obtained from fuzzy ideals and generalized fuzzy ideals of a double Boolean algebra D can entirely be determined by sets of fuzzy ideals and generalized fuzzy ideals of the Boolean algebra $D_{\sqcup}.$