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32 pages, 1142 KiB

Open AccessArticle

Fuzzy Graph Hyperoperations and Path-Based Algebraic Structures

by Antonios Kalampakas

Mathematics 2025, 13(13), 2180; https://doi.org/10.3390/math13132180 - 3 Jul 2025

Viewed by 345

This paper introduces a framework of hypercompositional algebra on fuzzy graphs by defining and analyzing fuzzy path-based hyperoperations. Building on the notion of strongest strong paths (paths that are both strength-optimal and composed exclusively of strong edges, where each edge achieves maximum connection strength between its endpoints), we define two operations: a vertex-based fuzzy path hyperoperation and an edge-based variant. These operations generalize classical graph hyperoperations to the fuzzy setting while maintaining compatibility with the underlying topology. We prove that the vertex fuzzy path hyperoperation is associative, forming a fuzzy hypersemigroup, and establish additional properties such as reflexivity and monotonicity with respect to

α

-cuts. Structural features such as fuzzy strong cut vertices and edges are examined, and a fuzzy distance function is introduced to quantify directional connectivity strength. We define an equivalence relation based on mutual full-strength reachability and construct a quotient fuzzy graph that reflects maximal closed substructures under the vertex fuzzy path hyperoperation. Applications are discussed in domains such as trust networks, biological systems, and uncertainty-aware communications. This work aims to lay the algebraic foundations for further exploration of fuzzy hyperstructures that support modeling, analysis, and decision-making in systems governed by partial and asymmetric relationships. Full article

(This article belongs to the Special Issue Advances in Hypercompositional Algebra and Its Fuzzifications)

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13 pages, 307 KiB

Open AccessArticle

On Kemeny Optimization Scheme for Fuzzy Set of Relations

by Serhii O. Mashchenko, Olena A. Kapustian and Bruno Rubino

Axioms 2023, 12(12), 1067; https://doi.org/10.3390/axioms12121067 - 21 Nov 2023

Cited by 1 | Viewed by 1059

Abstract

The present paper investigated the aggregation of individual preferences into a group fuzzy preference relation for a fuzzy set of decision-makers (DMs). This aggregation is based on the Kemeny optimization scheme. It was proven that this group relation is a Type-2 fuzzy relation (T2FR). The decomposition approach was used to analyze the group T2FR. It is shown that the group T2FR can be decomposed according to secondary membership grades into a finite collection of Type-1 fuzzy relations. Each of them is a group fuzzy relation for a crisp set of DMs, which is the corresponding

α

-cut of the original fuzzy set of DMs. Illustrative examples are given. Full article

(This article belongs to the Special Issue Stability, Approximation, Control and Application)

25 pages, 448 KiB

Open AccessArticle

Adjacent Vertex Distinguishing Coloring of Fuzzy Graphs

by Zengtai Gong and Chen Zhang

Mathematics 2023, 11(10), 2233; https://doi.org/10.3390/math11102233 - 10 May 2023

Cited by 2 | Viewed by 2386

Abstract

In this paper, we consider the adjacent vertex distinguishing proper edge coloring (for short, AVDPEC) and the adjacent vertex distinguishing total coloring (for short, AVDTC) of a fuzzy graph. Firstly, this paper describes the development process, the application areas, and the existing review research of fuzzy graphs and adjacent vertex distinguishing coloring of crisp graphs. Secondly, we briefly introduce the coloring theory of crisp graphs and the related theoretical basis of fuzzy graphs, and add some new classes of fuzzy graphs. Then, based on the

α

-cuts of fuzzy graphs and distance functions, we give two definitions of the AVDPEC of fuzzy graphs, respectively. A lower bound on the chromatic number of the AVDPEC of a fuzzy graph is obtained. With examples, we show that some results of the AVDPEC of a crisp graph do not carry over to our set up; the adjacent vertex distinguishing chromatic number of the fuzzy graph is different from the general chromatic number of a fuzzy graph. We also give a simple algorithm to construct a

(d, f)

-extended AVDPEC for fuzzy graphs. After that, in a similar way, two definitions of the AVDTC of fuzzy graphs are discussed. Finally, the future research directions of distinguishing coloring of fuzzy graphs are given. Full article

(This article belongs to the Special Issue Fuzzy Group Decision Making and Intelligent Systems: Recent Trends and Methodologies)

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19 pages, 3329 KiB

Open AccessArticle

An Intuitionistic Fuzzy-Rough Set-Based Classification for Anomaly Detection

by Fokrul Alom Mazarbhuiya and Mohamed Shenify

Appl. Sci. 2023, 13(9), 5578; https://doi.org/10.3390/app13095578 - 30 Apr 2023

Cited by 9 | Viewed by 2283

Abstract

The challenging issues of computer networks and databases are not only the intrusion detection but also the reduction of false positives and increase of detection rate. In any intrusion detection system, anomaly detection mainly focuses on modeling the normal behavior of the users and detecting the deviations from normal behavior, which are assumed to be potential intrusions or threats. Several techniques have already been successfully tried for this purpose. However, the normal and suspicious behaviors are hard to predict as there is no precise boundary differentiating one from another. Here, rough set theory and fuzzy set theory come into the picture. In this article, a hybrid approach consisting of rough set theory and intuitionistic fuzzy set theory is proposed for the detection of anomaly. The proposed approach is a classification approach which takes the advantages of both rough set and intuitionistic fuzzy set to deal with inherent uncertainty, vagueness, and indiscernibility in the dataset. The algorithm classifies the data instances in such a way that they can be expressed using natural language. A data instance can possibly or certainly belong to a class with degrees of membership and non-membership. The empirical study with a real-world and a synthetic dataset demonstrates that the proposed algorithm has normal true positive rates of 91.989% and 96.99% and attack true positive rates of 91.289% and 96.29%, respectively. Full article

(This article belongs to the Special Issue New Intrusion Detection Technology Driven by Artificial Intelligence)

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21 pages, 320 KiB

Open AccessArticle

A Certain Structure of Bipolar Fuzzy Subrings

by Hanan Alolaiyan, Muhammad Haris Mateen, Dragan Pamucar, Muhammad Khalid Mahmmod and Farrukh Arslan

Symmetry 2021, 13(8), 1397; https://doi.org/10.3390/sym13081397 - 1 Aug 2021

Cited by 8 | Viewed by 2656

Abstract

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and [...] Read more.

(α, β)

-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism. Full article

(This article belongs to the Special Issue Uncertain Multi-Criteria Optimization Problems II)

19 pages, 782 KiB

Open AccessArticle

A Novel Rough Set Model in Generalized Single Valued Neutrosophic Approximation Spaces and Its Application

by Zhi-Lian Guo, Yan-Ling Liu and Hai-Long Yang

Symmetry 2017, 9(7), 119; https://doi.org/10.3390/sym9070119 - 17 Jul 2017

Cited by 17 | Viewed by 4325

Abstract

In this paper, we extend the rough set model on two different universes in intuitionistic fuzzy approximation spaces to a single-valued neutrosophic environment. Firstly, based on the

(α, β, γ)

-cut relation [...] Read more.

In this paper, we extend the rough set model on two different universes in intuitionistic fuzzy approximation spaces to a single-valued neutrosophic environment. Firstly, based on the

(α, β, γ)

-cut relation

{\tilde{R}}_{{(α, β, γ)}}

, we propose a rough set model in generalized single-valued neutrosophic approximation spaces. Then, some properties of the new rough set model are discussed. Furthermore, we obtain two extended models of the new rough set model—the degree rough set model and the variable precision rough set model—and study some of their properties. Finally, we explore an example to illustrate the validity of the new rough set model. Full article

(This article belongs to the Special Issue Neutrosophic Theories Applied in Engineering)

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