The lattice of L-ideals of a ring is modular

Jahan, Iffat

doi:10.1016/j.fss.2011.12.012

Cited by 12 publications

(9 citation statements)

References 15 publications

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“…In [34], the author introduced the concept of L-fuzzy ideals of a commutative ring and gave some propositions (Propositions 2.4-2.6), where L is a completely distributive lattice. Here, we give the similar definition and results when L is a complete residuated lattice, which is a more general structure of truth values than a completely distributive lattice, and we omit the proof.…”

Section: Definition 21 ([27])mentioning

confidence: 99%

When do L-fuzzy ideals of a ring generate a distributive lattice?

Gao

2016

Open Mathematics

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Abstract:The notion of L-fuzzy extended ideals is introduced in a Boolean ring, and their essential properties are investigated. We also build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals. By defining an operator " " between two arbitrary L-fuzzy ideals in terms of L-fuzzy extended ideals, the result that "the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra" is immediately obtained. Furthermore, the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal, L-fuzzy extended ideals relative to an L-fuzzy subset, L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are studied in this paper.

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Section: Definition 21 ([27])mentioning

confidence: 99%

When do L-fuzzy ideals of a ring generate a distributive lattice?

Gao

2016

Open Mathematics

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“…Unfortunately after the emergence of metatheorem, not much attention has been paid on its further development or its application in other areas of algebra such as group theory and ring theory. There are only few papers related to this topic [3][4][5][6]. In [3], a new subdirect product theorem for fuzzy congruences is established.…”

Section: Introductionmentioning

confidence: 99%

“…Using this technique for constructing the join, A. Jain [5] has provided a much simpler and direct proof of modularity of the lattice of fuzzy normal subgroups. In the similar manner, I. Jahan [4] devised the join of two L-ideals by using the notion of tip extended pair of Lideals and established the modularity of the lattice of L-ideals of a ring which was an open problem. Here in this case the metatheorem or the subdirect product theorem were not applicable.…”

Section: Introductionmentioning

confidence: 99%

A New Metatheorem and Subdirect Product Theorem for L-Subgroups

Ajmal

Jahan

2018

Fuzzy Information and Engineering

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This paper is a continuation of the work of Tom Head 'Metatheorem for deriving fuzzy theorems from crisp versions'. The concept of natural extension is introduced which is then applied in the development of a new metatheorem in L-setting, where L is a complete chain. The application of this theorem is demonstrated through the notions of generated L-subgroups and commutator L-subgroups. Moreover, a new subdirect product theorem is developed wherein it is demonstrated that for a group G, the L-subgroup lattice can be represented as a subdirect product of copies of its associated lattice of crisp subgroups. The significance of this theorem is exhibited by applying it to a characterization of generalized cyclic groups in terms of the distributivity of the lattice of L-subgroups of a group.

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“…Tarnauceanu [20] characterized distributivity of the lattice of the fuzzy subgroups of a finite group. Recently in [11] the modularity of L-ideals of a ring is established where the subdirect product theorem of Tom Head does not apply.…”

Section: Introductionmentioning

confidence: 99%

The lattice of generalized normal L-subgroups

Bayrak

Yamak

2014

Journal of Intelligent &Amp; Fuzzy Systems

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Many studies have investigated the lattice structure of fuzzy substructures of algebraic sets such as group and ring. Some important results about modularity and distributivity have been obtained in these studies. In this paper, we first define the notion of (normal) (λ, µ)-L-subgroups on a group and investigate some of their properties. In particular, we discuss the relationships among ordinary L-subgroups, (∈, ∈ ∨q)-, (∈,∈ ∨q)-, (∈ λ , ∈ λ ∨q µ )-fuzzy subgroups and (λ, µ)-L-subgroups of a group. Also, we give a characterization of (normal) (λ, µ)-L-subgroup by means of (normal) L-subgroup. Finally, we discuss some properties of the lattices of normal (λ, µ)-L-subgroups and obtain that the lattice of all normal (0, µ)-L-subgroups of a group is modular. As consequence, we obtain that the lattices of all normal (∈, ∈ ∨q)-fuzzy subgroups and all normal L-subgroups of a group are modular.

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The lattice of L-ideals of a ring is modular

Cited by 12 publications

References 15 publications

When do L-fuzzy ideals of a ring generate a distributive lattice?

When do L-fuzzy ideals of a ring generate a distributive lattice?

A New Metatheorem and Subdirect Product Theorem for L-Subgroups

The lattice of generalized normal L-subgroups

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