title: On fuzzy ideals of BL-algebras.
abstract: In this paper we investigate further properties of fuzzy ideals of a BL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of a BL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime ideal ω is a fuzzy irreducible ideal if and only if ω(0) = 1 and |Im(ω)| = 2. We give the Krull-Stone representation theorem of fuzzy ideals in BL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of a BL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.
authors.affiliation: Department of Mathematics, Northwest University, Xi'an 710127, China ; College of Science, Xi'an University of Science and Technology, Xi'an 710054, China.
authors.forename: Biao Long
authors.identifier: 0000-0002-8447-6051
authors.initials: BL
authors.lastname: Meng
authors.affiliation: Department of Mathematics, Northwest University, Xi'an 710127, China.
authors.forename: Xiao Long
authors.initials: XL
authors.lastname: Xin
mesh_terms: D017143:Fuzzy Logic; D008962:Models, Theoretical