# On annihilators in BL-algebras

Yu Xi Zou; Xiao Long Xin; Peng Fei He

Open Mathematics (2016)

- Volume: 14, Issue: 1, page 324-337
- ISSN: 2391-5455

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topYu 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 -

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