On fuzzy ideals of fuzzy lattice

Mezzomo, Ivan; Bedregal, Benjamín R. C.; Santiago, Regivan H. N.

doi:10.1109/fuzz-ieee.2012.6251307

Cited by 7 publications

(5 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Mezzomo et al [25], Remark 3.1: "When A is fuzzy reflexive, then the fuzzy transitivity can be rewritten by replacing the "≥" by "=". In other words, A is fuzzy transitive iff A(x, z) = sup y∈X min{A(x, y), A(y, z)}, for all x, y, z ∈ X."…”

Section: Fuzzy Partial Order Relationsmentioning

confidence: 99%

“…Recently, both Fuzzy Ideals and Fuzzy Filters of a fuzzy lattice (X, A), were defined in the sense of Chon [9] as a crisp subset Y ⊆ X endowed with the restricted fuzzy order A| Y ×Y -see [25]. A discussion of such kind of ideals and filters as well as the investigation of their families was done in [26].…”

Section: Introductionmentioning

confidence: 99%

“…A discussion of such kind of ideals and filters as well as the investigation of their families was done in [26].…”

Section: Introductionmentioning

confidence: 99%

“…see [2,5,9,25,34]. The notion of Fuzzy Ideals arose in 1982, when Liu [23] defined fuzzy ideals of fuzzy invariant subgroups.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Types of fuzzy ideals in fuzzy lattices

Mezzomo

Bedregal

Santiago

2015

Journal of Intelligent &Amp; Fuzzy Systems

Self Cite

View full text Add to dashboard Cite

In this paper we consider the notion of Fuzzy Lattices, which was introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374). We propose some new notions for Fuzzy Ideals and Filters and provide a characterization of Fuzzy Ideals via α-level Sets and Support. Some types of ideals and filters, such as: Fuzzy Principal Ideals (Filters), Proper Fuzzy Ideals (Filters), Prime Fuzzy Ideals (Filters) and Fuzzy Maximal Ideals (Filters) are also provided. Some properties (analogous to the classical theory) are also proved and the notion of Homomorphism from fuzzy lattices as well as the demonstration of some important propositions about it are also provided.all α-cuts are sublattices of M. In 2009, Zhang, Xie and Fan [40] defined Fuzzy Complete Lattices as sets, X, endowed with a lattice-valued fuzzy order.More recently, Chon in [9], considering Zadeh's fuzzy orders [39], proposed a new notion for Fuzzy Lattices and studied the level sets of such structures, he also provided some results for Distributive and Modular Fuzzy Lattices.Although several different notions of fuzzy order relations have been given, for example see the references [6,8,13,15,36], Zadeh's notion [39] have been widely considered in recent years; e.g. see [2,5,9,25,34]. The notion of Fuzzy Ideals arose in 1982, when Liu [23] defined fuzzy ideals of fuzzy invariant subgroups. Since then, several papers have used it, for example: Majumdar and Sultana [24], and Navarro, Cortadellas and Robillo [29]. In 1990, Yuan and Wu [37] defined fuzzy ideal as a kind of fuzzy set under a conventional distributive lattice, and this approach has been followed by several authors, including Attallah [3], Koguep, Nkumi and Lele [20] and, more recently, by Davvaz . [25], Remark 3.2: "Since A is fuzzy antisymmetric, then the least upper (greatest lower) bound, if it exists, is unique." Example 2.1. Let X = {x, y, z, w} and let A : X × X −→ [0, 1] be a fuzzy relation such thatObserve that for Y = {z, w}, x, y and z are upper bounds of Y , but since A(z, w) = 0 and A(w, z) > 0, then LUB is z and the GLB is w.In the following, the reader can find the related tabular and graphical representations for A:As in the classical case, it is not true that every set of elements of a fuzzy poset has a least upper (greatest lower) bound. For example:Example 2.2. Let X = {x, y, z, w} and let A : X × X −→ [0, 1] be a fuzzy relation such that A(x, x) = A(y, y) = A(z, z) = A(w, w) = 1, A(x, y) = A(y, x) = A(x, z) = A(x, w) = A(y, z) = A(y, w) = A(z, w) = 0, A(z, x) = 0.5, A(w, x) = 0.8, A(z, y) = 0.2, A(w, y) = 0.4, and A(w, z) = 0.1. Then it is easily checked that A is a fuzzy partial order relation.According to Chon [9], Definition 3.2, a fuzzy poset (X, A) is a fuzzy lattice iff x ∨ y and x ∧ y exist for all x, y ∈ X. Moreover, for every fuzzy poset, (X, A), and Y ⊆ X. If B is A restricted to Y , B = A| Y ×Y , then (Y, B) is also a fuzzy poset.Remark 2.2. The Example 2.1 is an example of fuzzy lattice whereas the Example 2.2 is not. Definition 2.1. Let (X, A) be a fuzzy lattice. (Y, B...

show abstract

Section: Fuzzy Partial Order Relationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Types of fuzzy ideals in fuzzy lattices

Mezzomo

Bedregal

Santiago

2015

Journal of Intelligent &Amp; Fuzzy Systems

Self Cite

View full text Add to dashboard Cite

show abstract

“…Definition 25 ( [11]): Let (H,C) be a fuzzy lattice and I be any non empty subset of H. I is an ideal of (H,C), if the following two conditions hold true.…”

Section: (R A) Is Distributive If and Only Ifmentioning

confidence: 99%