A Notion of Convergence in Fuzzy Partially Ordered Sets

Georgiou, D.N.; Megaritis, A.C.; Prinos, G.A.

doi:10.3390/math8111958

Cited by 3 publications

(4 citation statements)

References 12 publications

(13 reference statements)

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“…Specifically, we examine the necessary and sufficient conditions that a fuzzy function ideal convergence class L, on a non-empty set X, should fulfill to determine a unique fuzzy topology δ on X such that I-convergence (L) coincides with I-convergence, with respect to δ. All the results obtained here are parallel to and extend those given in [18,19,21], for the ordinary topology, while simultaneously simplify the exposition and the underlying theory. In order to increase the utility of the present work, future research options may include the extension of the lattice background, L, to completely distributive lattices with an ordering-reversing involution as a tool to investigate the notion of L-fuzzy function ideal convergence and its applications in the more general context of L-fuzzy topological spaces.…”

Section: Discussionsupporting

confidence: 74%

“…We should note that previous ideas are reorganized efficiently and full proofs of the most important points are provided, rather than pointing out the necessary adaptations. In addition, this work extends the results of [21] to fuzzy topological spaces.…”

Section: Introductionsupporting

confidence: 61%

“…Subsequently, in [19], they provided similar results by considering arbitrary ideals. In continuation, in [21] they extended and simplified the results of the last two papers by considering functions instead of nets and a smaller set of axioms, to be fulfilled by class C, in order for the last to be topological.…”

Section: Introductionmentioning

confidence: 99%

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A New Notion of Fuzzy Function Ideal Convergence

Georgiou,

Prinos

2024

Mathematics

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P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class, L, on a non-empty set, X, consisting of triads (f,e,I), where f is a function from a non-empty set, D, to the set FP(X) of fuzzy points in X, which we call fuzzy function, e∈FP(X), and I is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology, δ, on X, such that (f,e,I)∈L iff fI-converges to e, relative to the fuzzy topology δ.

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Section: Discussionsupporting

confidence: 74%

Section: Introductionsupporting

confidence: 61%

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A New Notion of Fuzzy Function Ideal Convergence

Georgiou,

Prinos

2024

Mathematics

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“…Then, Venugopalan [2] developed a structure of fuzzy ordered sets. Since then, many authors have studied fuzzy relations and ordering by using different approaches [3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

A Study on Fuzzy Order Bounded Linear Operators in Fuzzy Riesz Spaces

et al. 2021

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This paper aims to study fuzzy order bounded linear operators between two fuzzy Riesz spaces. Two lattice operations are defined to make the set of all bounded linear operators as a fuzzy Riesz space when the codomain is fuzzy Dedekind complete. As a special case, separation property in fuzzy order dual is studied. Furthermore, we studied fuzzy norms compatible with fuzzy ordering (fuzzy norm Riesz space) and discussed the relation between the fuzzy order dual and topological dual of a locally convex solid fuzzy Riesz space.

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Statistical and Ideal Convergences in Topology

2023

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The notion of convergence wins its own important part in the branch of Topology. Convergences in metric spaces, topological spaces, fuzzy topological spaces, fuzzy metric spaces, partially ordered sets (in short, posets), and fuzzy ordered sets (in short, fosets) develop significant chapters that attract the interest of many studies. In particular, statistical and ideal convergences play their own important role in all these areas. A lot of studies have been devoted to these special convergences, and many results have been proven. As a consequence, the necessity to produce and extend new results arises. Since there are many results on different kinds of convergences in different areas, we present a review paper on this research topic in order to collect the most essential results, which leads us to provide open questions for further investigation. More precisely, we present and gather definitions and results which have been proven for different kinds of convergences, mainly on statistical/ideal convergences, in metric spaces, topological spaces, fuzzy topological spaces, fuzzy metric spaces, posets, and fosets. Based on this presentation, we provide new open problems for further investigation on related topics. The study of these problems will create new “roads”, enriching the branch of convergences in the field of Topology.

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A Notion of Convergence in Fuzzy Partially Ordered Sets

Cited by 3 publications

References 12 publications

A New Notion of Fuzzy Function Ideal Convergence

A New Notion of Fuzzy Function Ideal Convergence

A Study on Fuzzy Order Bounded Linear Operators in Fuzzy Riesz Spaces

Statistical and Ideal Convergences in Topology

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