Fuzzy equivalence relations and their equivalence classes - ScienceDirect

Fuzzy Sets and Systems

Volume 158, Issue 12, 16 June 2007, Pages 1295-1313

Fuzzy Sets and Systems

https://doi.org/10.1016/j.fss.2007.01.010 Get rights and content

Abstract

In this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice. We give certain characterizations of fuzzy semi-partitions and fuzzy partitions over a complete residuated lattice, as well as over a linearly ordered complete Heyting algebra. In the latter case, for a fuzzy equivalence relation over a linearly ordered complete Heyting algebra, we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it. Most of the presented results are new, but some of them are generalizations of known results given in a way which simplifies and clarifies them.

References (34)

D. Boixader et al.
Transitive closure and betweenness relations
Fuzzy Sets and Systems
(2001)
D. Butnariu
Additive fuzzy measures and integrals
J. Math. Anal. Appl.
(1983)
B. De Baets et al.
The construction of possibility measures from samples on T-semi-partitions
Inform. Sci.
(1998)
B. De Baets et al.
T-partitions of the real line generated by idempotent shapes
Fuzzy Sets and Systems
(1997)
B. De Baets et al.
$T$ -partitions
Fuzzy Sets and Systems
(1998)
B. De Baets et al.
Metrics and $T$ -equalities
J. Math. Anal. Appl.
(2002)
M. Demirci et al.
Fuzzy groups, fuzzy functions and fuzzy equivalence relations
Fuzzy Sets and Systems
(2004)
J.-P. Doignon et al.
Dimension of valued relations
European J. Oper. Res.
(2000)
U. Höhle
On the fundamentals of fuzzy set theory
J. Math. Anal. Appl.
(1996)
J. Jacas
Similarity relations. The calculation of minimal generating families
Fuzzy Sets and Systems
(1990)

J. Jacas et al.

Fuzzy T-transitive relations: eigenvectors and generators

Fuzzy Sets and Systems

(1995)

B. Moser et al.

A relationship between equality relations and the T-redundancy of fuzzy partitions and its applications to Sugeno controllers

Fuzzy Sets and Systems

(2000)

V. Murali

Fuzzy equivalence relations

Fuzzy Sets and Systems

(1989)

S. Ovchinnikov

Similarity relations, fuzzy partitions, and fuzzy orderings

Fuzzy Sets and Systems

(1991)

E. Ruspini

A new approach to clustering

Inform. Control

(1969)

L. Valverde

On the structure of F-indistinguishability operators

Fuzzy Sets and Systems

(1985)

L.A. Zadeh

Similarity relations and fuzzy orderings

Inform. Sci.

(1971)

Cited by (56)

Clustering sustainable suppliers in the plastics industry: A fuzzy equivalence relation approach
2023, Journal of Environmental Management
Nowadays, pure economic supply chain management is not commonly contemplated among companies (especially buyers), as recently novel dimensions of supply chains, e.g., environmental, sustainability, and risk, play significant roles. In addition, since companies prefer buying their needs from a group of suppliers, the problem of supplier selection is not solely choosing or qualifying a supplier from among others. Buyers, hence, commonly assemble a portfolio of suppliers by looking at the multi-dimensional pre-determined selection criteria. Since sustainable supplier selection criteria are often assessed by linguistic terms, an appropriate clustering approach is required. This paper presents an innovative way to implement fuzzy equivalence relation to clustering sustainable suppliers through developing a comprehensive taxonomy of sustainable supplier selection criteria, including supply chain risk. Fifteen experts participated in this study to evaluate 20 suppliers and cluster them in the plastics industry. Findings reveal that the best partitioning occurs when the suppliers are divided into two clusters, with 4 (20%) and 16 (80%) suppliers, respectively. The four suppliers in cluster one are performing better in terms of the capability of supplier/delivery, service, risk, and sustainability criteria such as environment protection/management, and green innovation. These factors are critical in clustering and selecting sustainable suppliers. The originality of this study lies in developing an all-inclusive set of criteria for clustering sustainable suppliers and adding risk factors to the conventional supplier selection criteria. In addition to partitioning the suppliers and determining the best-performing ones, this study also highlights the most influential factors by analysing the suppliers in the best cluster.
Computing the fuzzy partition corresponding to the greatest fuzzy auto-bisimulation of a fuzzy graph-based structure under the Gödel semantics
2023, Information Sciences
The similarity measure based on fuzzy bisimulation has the Hennessy-Milner property as a strong logical foundation. It is useful for classification and clustering. In this work, we design an efficient algorithm with the complexity $O ((m \log l + n) \log n)$ for computing the fuzzy partition corresponding to the greatest fuzzy auto-bisimulation of a finite fuzzy labeled graph G under the Gödel semantics, where n, m and l are the number of vertices, the number of non-zero edges and the number of different fuzzy degrees of edges of G, respectively. Our notion of fuzzy partition is novel, defined for finite sets with respect to the Gödel t-norm, with the aim to facilitate the computation of the greatest fuzzy auto-bisimulation. By using that algorithm, we also provide an algorithm with the complexity $O (m \cdot \log l \cdot \log n + n^{2})$ for computing the greatest fuzzy bisimulation between two finite fuzzy labeled graphs under the Gödel semantics. This latter algorithm is better (has a lower complexity order) than the previously known algorithms for the considered problem. Our algorithms can be restated for other fuzzy graph-based structures such as fuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models, fuzzy social networks and fuzzy interpretations in fuzzy description logics.
Approximate positional analysis of fuzzy social networks
2023, Fuzzy Sets and Systems
In order to facilitate the processing of vast amounts of data that emerges in the study of fuzzy social networks, scholars have developed various procedures for reducing these networks. Some procedures use regular and structural fuzzy relations to reduce such networks. In this article, we generalize the notions of regular and structural fuzzy relations to obtain even better reductions of fuzzy networks. More precisely, for a fuzzy social network given by the set of social entities and the family of fuzzy relations between them, we define μ-approximate regular fuzzy relations, where μ is the degree taken from the underlying set of truth values, which is a complete Heyting algebra. Using these specific fuzzy relations, we show that it is possible to reduce a fuzzy social network in some cases when previously developed algorithms fail to reduce it. We investigate the properties of μ-approximate regular fuzzy relations. We show that the blockmodel of a fuzzy social network, which is its reduced fuzzy social network built via μ-approximate regular fuzzy preorder, retains specific structure-preserving properties. We give a method for calculating the greatest μ-approximate regular fuzzy relation on a given fuzzy social network. For fuzzy social networks defined over the real-unit interval $[0, 1]$ , we give a procedure that determines all subintervals of $[0, 1]$ that share the greatest μ-approximate regular fuzzy relation. Analogous results are provided for μ-approximate structural fuzzy relations.
On the solvability of weakly linear systems of fuzzy relation equations<sup>☆</sup>
2022, Information Sciences
Systems of fuzzy relation equations and inequalities in which an unknown fuzzy relation is on the one side of the equation or inequality are linear systems. They are the most studied ones, and a vast literature on linear systems focuses on finding solutions and solvability criteria for such systems. The situation is quite different with the so-called weakly linear systems, in which an unknown fuzzy relation is on both sides of the equation or inequality. Precisely, the scholars have only given the characterization of the set of exact solutions to such systems. This paper describes the set of fuzzy relations that solve weakly linear systems to a certain degree and provides ways to compute them. We pay special attention to developing the algorithms for computing fuzzy preorders and fuzzy equivalences that are solutions to some extent to weakly linear systems. We establish additional properties for the set of such approximate solutions over some particular types of complete residuated lattices. We demonstrate the advantage of this approach via many examples that arise from the problem of aggregation of fuzzy networks.
Symmetric difference operators based on thresholds
2020, Fuzzy Sets and Systems
Here, we study two formulas that define the symmetric difference operators of fuzzy sets. In fuzzy logic, we call this an exclusive or operator. An interesting open problem was to find a certain class of operator systems where the two well-known definitions of the symmetric difference operator are equivalent, and it is associative. Here, we state a functional equation which characterizes the equivalence of these two forms. We present a general class of operators which fulfills the associativity requirement and we show that the two definitions are equivalent. We generalize the symmetric difference operator to the n-variable case; and based on the latter the weighted form can be obtained. We will give concrete forms of these operators and we will also show that they have different types of transitivity properties.
Computation of the greatest right and left invariant fuzzy quasi-orders and fuzzy equivalences
2018, Fuzzy Sets and Systems
Citation Excerpt :
The concept of a fuzzy relation as a generalization of the ordinary (crisp) relation was introduced by Zadeh in his seminal paper on fuzzy sets [52]. Later, concepts of a fuzzy equivalence and fuzzy quasi-order, as generalizations of the ordinary (crisp) equivalence and quasi-order, were studied by numerous authors, see for example [10,17,41,42,53]. In this paper we study special types of fuzzy equivalences and fuzzy quasi-orders, namely, right and left invariant fuzzy quasi-orders, as well as right and left invariant fuzzy equivalences.
Right invariant fuzzy quasi-orders for fuzzy automata are broadly studied in the recent literature, as they arise as solutions to particular systems of fuzzy relation equations and inequalities. Some of their applications include determinization and state reduction procedures, as well as simulations and bisimulations for fuzzy automata. In this paper we provide a procedure for computing the greatest right invariant fuzzy quasi-order for a given fuzzy automaton over a complete residuated lattice. The proposed procedure terminates in a finite number of steps whenever the underlying structure of the fuzzy automaton is locally finite. When the previous condition is not satisfied, we show that the greatest right invariant fuzzy quasi-order can be obtained by taking the limit value of the convergent array of fuzzy quasi-orders for fuzzy automata over BL-algebras on the real unit interval $[0, 1]$ . Analogous procedures for computing the greatest left invariant fuzzy quasi-order, as well as the greatest right and left invariant fuzzy equivalences for a fuzzy automaton are also presented. In addition, the faster algorithm for computing the greatest right invariant equivalence on a nondeterministic automaton is also presented.

View all citing articles on Scopus

^☆: Research supported by Ministry of Science and Environmental Protection, Republic of Serbia, Grant No. 144011.

Copyright © 2007 Elsevier B.V. All rights reserved.

Notes on topological BL-algebras
Fuzzy Sets and Systems, Volume 350, 2018, pp. 33-40
Jiang Yang, …, Peng Fei He
Building Taxonomy for developing strategic partnerships with Original Equipment Manufacturers of a firm
Materials Today: Proceedings, Volume 5, Issue 11, Part 3, 2018, pp. 25541-25552
Saad Parvez, G.A. Harmain
Some properties of fuzzy dual cones
Fuzzy Sets and Systems, Volume 331, 2018, pp. 78-84
O.V. Baskov
Tunable equivalence fuzzy associative memories
Fuzzy Sets and Systems, Volume 292, 2016, pp. 242-260
Estevão Esmi, …, Sandra Sandri
Relations arising from coverings and their topological structures
International Journal of Approximate Reasoning, Volume 80, 2017, pp. 348-358
Guilong Liu, …, Jiyang Zou
Tense operators in fuzzy logic
Fuzzy Sets and Systems, Volume 276, 2015, pp. 100-113
Ivan Chajda, Jan Paseka