‎Lattices of (Generalized) Fuzzy Ideals in Double Boolean Algebras

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 D := (D; ⊓; ⊔; :; ⌟;?;⊤) 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⊔.