MV-algebras derived from ideals in BL-algebras

Lele, Celestin; Nganou, Jean B.

doi:10.1016/j.fss.2012.09.014

Cited by 53 publications

(20 citation statements)

References 12 publications

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“…We note that in MV-algebras the ideals and filters are dual. In 2013, Lele and Nganou [5] constructed some examples to show that, unlike in MV-algebras, ideals and filters are dual but behave differently in BL-algebras. In 2014, in the framework of MV-algebras, Forouzesh et al [6] published a study on obstinate ideals; they investigated some relationships between the obstinate ideals and the other ideals of an MV-algebra.…”

Section: Introductionmentioning

confidence: 99%

The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices

Holdon

2020

Open Mathematics

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In this paper, by using the ideal theory in residuated lattices, we construct the prime and maximal spectra (Zariski topology), proving that the prime and maximal spectra are compact topological spaces, and in the case of De Morgan residuated lattices they become compact {T}_{0} topological spaces. At the same time, we define and study the reticulation functor between De Morgan residuated lattices and bounded distributive lattices. Moreover, we study the I-topology (I comes from ideal) and the stable topology and we define the concept of pure ideal. We conclude that the I-topology is in fact the restriction of Zariski topology to the lattice of ideals, but we use it for simplicity. Finally, based on pure ideals, we define the normal De Morgan residuated lattice (L is normal iff every proper ideal of L is a pure ideal) and we offer some characterizations.

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Section: Introductionmentioning

confidence: 99%

The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices

Holdon

2020

Open Mathematics

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“…So the notion of ideals is missing in BL-algebras. To fill this gap the paper [3] introduced the notion of ideals in BL-algebras, which…”

Section: Introductionmentioning

confidence: 99%

On annihilators in BL-algebras

Zou

Xin

2016

Open Mathematics

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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/; Â/ 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/I \; _ An.L/ ; ?; f0g; 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 is the relative pseudo-complement of J with respect to I in the ideal lattice .I.L/; Â/. 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˛-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.

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“…In the meantime, several authors have claimed in recent works that the notion of ideals is missing in BL-algebras. Lele [7], [8] studied the notion of fuzzy n-fold (positive) implicative filter and fuzzy n-fold obstinate filter in BL-algebras. In 2012, Borzooei and Paad [12], introduced the notions of n-fold integral filter and nfold integral BL algebra.…”

Section: Introductionmentioning

confidence: 99%

“…The notion of (fuzzy) ideal has been introduced in many algebraic structures such as lattices, rings, MV -algebras. Zhang et al [16] studied the notion of fuzzy ideals in BL-algebras and in 2013, Lele [7], introduced the notion of Boolean ideals and analyzed the relationship between ideals and filters by using the set of complement elements. Now, in this paper, we introduce concepts of fuzzy n-fold integral ideals in BL-algebras and we state and prove several property of fuzzy n-fold integral ideals.…”

Section: Introductionmentioning

confidence: 99%