Research

Open AccessArticle

First Entire Zagreb Index of Fuzzy Graph and Its Application

by

Umapada Jana

and

Ganesh Ghorai

Axioms 2023, 12(5), 415; https://doi.org/10.3390/axioms12050415 - 24 Apr 2023

Viewed by 426

Abstract

The first entire Zagreb index (FEZI) is a graph parameter that has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this research paper, the FEZI was explored for a variety of fuzzy graphs, [...] Read more.

The first entire Zagreb index (FEZI) is a graph parameter that has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this research paper, the FEZI was explored for a variety of fuzzy graphs, including star, firefly graph, cycle, path, fuzzy subgraph, vertex elimination, and edge elimination. This study presented several results, including determining the relationship between two isomorphic fuzzy graphs and between a path and cycle (connecting both end vertices of the path). This research also deals with the analysis of

α

-cut fuzzy graphs and establishes bounds for some fuzzy graphs. To apply these findings to modern life problems, the research team utilized the results to identify areas that require more development in internet systems. These results have practical implications for enhancing the efficiency and effectiveness of internet systems. The conclusion drawn from this research can be used to inform future research and aid in the development of more efficient and effective systems in various fields. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

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Open AccessArticle

On Two Intuitionistic Fuzzy Modal Topological Structures

by

Krassimir Atanassov

,

Nora Angelova

and

Tania Pencheva

Axioms 2023, 12(5), 408; https://doi.org/10.3390/axioms12050408 - 22 Apr 2023

Viewed by 338

Abstract

The concept of an Intuitionistic Fuzzy Modal Topological Structure (IFMTS) was introduced previously, and some of its properties were studied. So far, there are two different IFMTSs based on the classical intuitionistic fuzzy operations: “union” (∪) and “intersection” (∩). In the present paper, [...] Read more.

The concept of an Intuitionistic Fuzzy Modal Topological Structure (IFMTS) was introduced previously, and some of its properties were studied. So far, there are two different IFMTSs based on the classical intuitionistic fuzzy operations: “union” (∪) and “intersection” (∩). In the present paper, two new IFMTSs are developed. They are based on new intuitionistic fuzzy topological operators from closure and interior types, introduced here for the first time, and on the two standard intuitionistic fuzzy modal operators □ and ◊. Some basic properties of the new IFMTSs are discussed. The newly presented IFMTSs could be considered as a basis for the next research on the IFMTSs. Some ideas for the future development of the IFMTS theory and open problems are formulated, related to the existence of other intuitionistic fuzzy operations that can generate new intuitionistic fuzzy topological operators and, respectively, new IFMTSs. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

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Open AccessArticle

Fuzzy Algebras of Concepts

by

Manuel Ojeda-Hernández

,

Domingo López-Rodríguez

and

Pablo Cordero

Axioms 2023, 12(4), 324; https://doi.org/10.3390/axioms12040324 - 26 Mar 2023

Viewed by 411

Abstract

Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, [...] Read more.

Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, is to present a fuzzy powerset lattice gradation, that coincides with the preconcept lattice at its 1-cut. The second and more significant contribution, is to introduce a preconcept algebra gradation that yields different algebras for protoconcepts, semiconcepts, and concepts at different cuts. This result reveals new insights into the structure and properties of the different categories of preconcepts. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

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Open AccessArticle

Partial Residuated Implications Induced by Partial Triangular Norms and Partial Residuated Lattices

by

Xiaohong Zhang

,

Nan Sheng

and

Rajab Ali Borzooei

Axioms 2023, 12(1), 63; https://doi.org/10.3390/axioms12010063 - 06 Jan 2023

Cited by 1 | Viewed by 808

Abstract

This paper reveals some relations between fuzzy logic and quantum logic on partial residuated implications (PRIs) induced by partial t-norms as well as proposes partial residuated monoids (PRMs) and partial residuated lattices (PRLs) by defining partial adjoint pairs. First of all, we introduce [...] Read more.

This paper reveals some relations between fuzzy logic and quantum logic on partial residuated implications (PRIs) induced by partial t-norms as well as proposes partial residuated monoids (PRMs) and partial residuated lattices (PRLs) by defining partial adjoint pairs. First of all, we introduce the connection between lattice effect algebra and partial t-norms according to the concept of partial t-norms given by Borzooei, together with the proof that partial operation in any commutative quasiresiduated lattice is partial t-norm. Then, we offer the general form of PRI and the definition of partial fuzzy implication (PFI), give the condition that partial residuated implication is a fuzzy implication, and prove that each PRI is a PFI. Next, we propose PRLs, study their basic characteristics, discuss the correspondence between PRLs and lattice effect algebras (LEAs), and point out the relationship between LEAs and residuated partial algebras. In addition, like the definition of partial t-norms, we provide the notions of partial triangular conorms (partial t-conorms) and corresponding partial co-residuated lattices (PcRLs). Lastly, based on partial residuated lattices, we define well partial residuated lattices (wPRLs), study the filter of well partial residuated lattices, and then construct quotient structure of PRMs. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

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Open AccessArticle

The Single Axiomatization on CCRL-Fuzzy Rough Approximation Operators and Related Fuzzy Topology

by

Yaoliang Xu

,

Dandan Zou

and

Lingqiang Li

Axioms 2023, 12(1), 37; https://doi.org/10.3390/axioms12010037 - 28 Dec 2022

Viewed by 621

Abstract

It is well known that lattice-valued rough sets are important branches of fuzzy rough sets. The axiomatic characterization and related topology are the main research directions of lattice-valued rough sets. For

L = (L, ⊛)

, a complete co-residuated lattice [...] Read more.

It is well known that lattice-valued rough sets are important branches of fuzzy rough sets. The axiomatic characterization and related topology are the main research directions of lattice-valued rough sets. For

L = (L, ⊛)

, a complete co-residuated lattice (CCRL), Qiao recently defined an L-fuzzy lower approximation operator (

L F L A O

) on the basis of the L-fuzzy relation. In this article, we give a further study on Qiao’s

L F L A O

around the axiomatic characterization and induced L-topology. Firstly, we investigate and discuss three new

L F L A O

generated by ⊛-transitive, ⊛-Euclidean and ⊛-mediated L-fuzzy relations. Secondly, we utilize a single axiom to characterize the

L F L A O

generated by serial, symmetric, reflexive, ⊛-transitive and ⊛-mediate L-fuzzy relations and their compositions. Thirdly, we present a method to generate Alexandrov L-topology (

A L T P O

) from

L F L A O

and construct a bijection between

A L T P O

and ⊛-preorder (i.e., reflexive and ⊛-transitive L-fuzzy relation) on the same underlying set. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

Open AccessArticle

Pseudo Overlap Functions, Fuzzy Implications and Pseudo Grouping Functions with Applications

by

,

,

,

,

,

and

Axioms 2022, 11(11), 593; https://doi.org/10.3390/axioms11110593 - 26 Oct 2022

Cited by 6 | Viewed by 743

Abstract

Overlap and grouping functions are important aggregation operators, especially in information fusion, classification and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we find overlap functions as well as [...] Read more.

Overlap and grouping functions are important aggregation operators, especially in information fusion, classification and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we find overlap functions as well as grouping functions are required to be commutative (or symmetric), which limit their wide applications. For the above reasons, this paper expands the original notions of overlap functions and grouping functions, and the new concepts of pseudo overlap functions and pseudo grouping functions are proposed on the basis of removing the commutativity of the original functions. Some examples and construction methods of pseudo overlap functions and pseudo grouping functions are presented, and the residuated implication (co-implication) operators derived from them are investigated. Not only that, some applications of pseudo overlap (grouping) functions in multi-attribute (group) decision-making, fuzzy mathematical morphology and image processing are discussed. Experimental results show that, in many application fields, pseudo overlap functions and pseudo grouping functions have greater flexibility and practicability. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

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Open AccessArticle

Three-Way Fuzzy Sets and Their Applications (II)

by

Jingqian Wang

,

Xiaohong Zhang

and

Qingqing Hu

Axioms 2022, 11(10), 532; https://doi.org/10.3390/axioms11100532 - 05 Oct 2022

Cited by 11 | Viewed by 807

Abstract

Recently, the notion of a three-way fuzzy set is presented, inspired by the basic ideas of three-way decision and various generalized fuzzy sets, including lattice-valued fuzzy sets, partial fuzzy sets, intuitionistic fuzzy sets, etc. As the new theory of uncertainty, it has been [...] Read more.

Recently, the notion of a three-way fuzzy set is presented, inspired by the basic ideas of three-way decision and various generalized fuzzy sets, including lattice-valued fuzzy sets, partial fuzzy sets, intuitionistic fuzzy sets, etc. As the new theory of uncertainty, it has been used in attribute reduction and as a new control method for the water level. However, as an extension of a three-way decision, this new theory has not been used in multi-criteria decision making (MCDM for short). Based on the previous work, in this paper, we present rough set models based on three-way fuzzy sets, which extend the existing fuzzy rough set models in both complete and incomplete information systems. Furthermore, the new models are used to solve the issue of MCDM. Firstly, three-way fuzzy relation rough set and three-way fuzzy covering rough set models are presented for complete and incomplete information systems. Because almost all existing fuzzy rough set models are proposed under complete information, the new proposed models can be seen as a supplement to these existing models. Then, a relationship between the three-way fuzzy relation rough set and the three-way fuzzy covering rough set is presented. Finally, a novel method for the issue of MCDM is presented under the novel three-way fuzzy rough set models, which is used in paper defect diagnosis. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

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Open AccessArticle

Comparison of Overlap and Grouping Functions

by

Songsong Dai

Axioms 2022, 11(8), 420; https://doi.org/10.3390/axioms11080420 - 20 Aug 2022

Cited by 2 | Viewed by 610

Abstract

This paper investigates the pointwise comparability of overlap and grouping functions which obtained by Bustince et al.’s and Bedregal et al.’s generator pairs, respectively. Some necessary and sufficient conditions for the comparison of these functions are proved. We also introduce some compositions of [...] Read more.

This paper investigates the pointwise comparability of overlap and grouping functions which obtained by Bustince et al.’s and Bedregal et al.’s generator pairs, respectively. Some necessary and sufficient conditions for the comparison of these functions are proved. We also introduce some compositions of these functions and study the order preservation of these compositions. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

Open AccessArticle

Transposition Regular TA-Groupoids and Their Structures

by

Xiaogang An

and

Xiaohong Zhang

Axioms 2022, 11(8), 378; https://doi.org/10.3390/axioms11080378 - 30 Jul 2022

Viewed by 987

Abstract

Tarski associative groupoid (TA-groupoid) is a kind of non-associative groupoid satisfying Tarski associative law. In this paper, the new notions of transposition regular TA-groupoid are proposed and their properties and structural characteristics are studied by using band and quasi-separativity. In particular, the following [...] Read more.

Tarski associative groupoid (TA-groupoid) is a kind of non-associative groupoid satisfying Tarski associative law. In this paper, the new notions of transposition regular TA-groupoid are proposed and their properties and structural characteristics are studied by using band and quasi-separativity. In particular, the following conclusions are strictly proved: (1) every left transposition regular TA-groupoid is a semigroup; (2) every left transposition regular TA-groupoid is the disjoint union of sub Abelian groups; and (3) a finite TA-groupoid with quasi-separativity and a finite left transposition regular TA-groupoid are equivalent. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

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Open AccessArticle

On Implicative Derivations of MTL-Algebras

by

,

,

and

Axioms 2022, 11(7), 340; https://doi.org/10.3390/axioms11070340 - 15 Jul 2022

Cited by 1 | Viewed by 686

Abstract

This paper introduces the implicative derivations and gives some of their characterizations on MTL-algebras. Furthermore, we provide some representation of MTL-algebras by implicative derivations and obtain some representation of Boolean algebra via the algebra of all implicative derivations. Finally, we explore the relationship [...] Read more.

This paper introduces the implicative derivations and gives some of their characterizations on MTL-algebras. Furthermore, we provide some representation of MTL-algebras by implicative derivations and obtain some representation of Boolean algebra via the algebra of all implicative derivations. Finally, we explore the relationship between implicative derivation and other operators on MTL-algebras and show that there exists a bijection between the sets of multiplier and implicative derivations on IMTL-algebras. The results of this paper can provide the common properties of implicative derivations in the t-norm-based fuzzy logical algebras. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

Open AccessArticle

On r-Ideals and m-k-Ideals in BN-Algebras

by

,

,

,

and

Axioms 2022, 11(6), 268; https://doi.org/10.3390/axioms11060268 - 02 Jun 2022

Viewed by 1212

Abstract

A BN-algebra is a non-empty set

X

with a binary operation “

*

” and a constant 0 that satisfies the following axioms:

(B 1) x * x = 0,

[...] Read more.

A BN-algebra is a non-empty set

X

with a binary operation “

*

” and a constant 0 that satisfies the following axioms:

(B 1) x * x = 0,

(B 2) x * 0 = x,

and

(B N) (x * y) * z = (0 * z) * (y * x)

for all

x, y, z \in X

. A non-empty subset

I

of

X

is called an ideal in BN-algebra X if it satisfies

0 \in X

and if

y \in I

and

x * y \in I

, then

x \in I

for all

x, y \in X

. In this paper, we define several new ideal types in BN-algebras, namely, r-ideal, k-ideal, and m-k-ideal. Furthermore, some of their properties are constructed. Then, the relationships between ideals in BN-algebra with r-ideal, k-ideal, and m-k-ideal properties are investigated. Finally, the concept of r-ideal homomorphisms is discussed in BN-algebra. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

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