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On annihilators in BL-algebras

2016; De Gruyter Open; Volume: 14; Issue: 1 Linguagem: Inglês

10.1515/math-2016-0029

2391-5455

Yu-Xi Zou, Xiao Long Xin, Peng He,

Fuzzy and Soft Set Theory

Abstract In the paper, we introduce the notion of annihilators in BL-algebras and investigate some related properties of them. We get that the ideal lattice ( I ( L ), ⊆) is pseudo-complemented, and for any ideal I , its pseudo-complement is the annihilator I ⊥ of I . Also, we define the An ( L ) to be the set of all annihilators of L , then we have that ( An (L); ⋂,∧ An ( L ) ,⊥,{0}, L ) is a Boolean algebra. In addition, we introduce the annihilators of a nonempty subset X of L with respect to an ideal I and study some properties of them. As an application, we show that if I and J are ideals in a BL-algebra L , then J I ⊥ $J_I^ \bot $ is the relative pseudo-complement of J with respect to I in the ideal lattice ( I ( L ), ⊆). Moreover, we get some properties of the homomorphism image of annihilators, and also give the necessary and sufficient condition of the homomorphism image and the homomorphism pre-image of an annihilator to be an annihilator. Finally, we introduce the notion of α -ideal and give a notation E ( I ). We show that ( E ( I ( L )), ∧ E , ∨ E , E (0) , E ( L ) is a pseudo-complemented lattice, a complete Brouwerian lattice and an algebraic lattice, when L is a BL-chain or a finite product of BL-chains.