On annihilators in BL-algebras

Yu Xi Zou, Xiao Long Xin, and Peng Fei He. "On annihilators in BL-algebras." Open Mathematics 14.1 (2016): 324-337. <http://eudml.org/doc/277099>.

@article{YuXiZou2016,
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 [...] JI⊥$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.},
author = {Yu Xi Zou, Xiao Long Xin, Peng Fei He},
journal = {Open Mathematics},
keywords = {BL-algebra; MV-algebra; Ideal; Annihilator; Homomorphism; ideal; annihilator; homomorphism},
language = {eng},
number = {1},
pages = {324-337},
title = {On annihilators in BL-algebras},
url = {http://eudml.org/doc/277099},
volume = {14},
year = {2016},
}

TY - JOUR
AU - Yu Xi Zou
AU - Xiao Long Xin
AU - Peng Fei He
TI - On annihilators in BL-algebras
JO - Open Mathematics
PY - 2016
VL - 14
IS - 1
SP - 324
EP - 337
AB - 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 [...] JI⊥$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.
LA - eng
KW - BL-algebra; MV-algebra; Ideal; Annihilator; Homomorphism; ideal; annihilator; homomorphism
UR - http://eudml.org/doc/277099
ER -