Meng Biao Long, Xin Xiao Long
Department of Mathematics, Northwest University, Xi'an 710127, China ; College of Science, Xi'an University of Science and Technology, Xi'an 710054, China.
Department of Mathematics, Northwest University, Xi'an 710127, China.
ScientificWorldJournal. 2014;2014:757382. doi: 10.1155/2014/757382. Epub 2014 Apr 17.
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.
在本文中，我们进一步研究了BL - 代数的模糊理想的性质。引入了BL - 代数的模糊素理想、模糊不可约理想和模糊哥德尔理想的概念，并研究了它们的一些性质。我们给出了一种由模糊集生成模糊理想的方法。我们证明了每个模糊不可约理想都是模糊素理想，但模糊素理想不一定是模糊不可约理想，并证明了模糊素理想ω是模糊不可约理想当且仅当ω(0) = 1且|Im(ω)| = 2。我们给出了BL - 代数中模糊理想的克鲁尔 - 斯通表示定理。此外，我们证明了BL - 代数的所有模糊理想构成的格是一个完全分配格。最后，证明了每个模糊布尔理想都是模糊哥德尔理想，但反之不成立。
