Elsevier

Fuzzy Sets and Systems

Volume 360, 1 April 2019, Pages 1-32
Fuzzy Sets and Systems

Some results on the degree of symmetry of fuzzy relations

https://doi.org/10.1016/j.fss.2018.05.020Get rights and content

Abstract

In this paper, we investigate the degree of the symmetry of fuzzy relations on a set X. Based on the fuzzy e-equality derived from uninorms with unit element e, the degree of the symmetry of fuzzy relations are defined by two different approaches, the type-I degree of symmetry and the type-II degree of symmetry. The main work of this paper include: first, we discuss some basic properties of those two degrees, especially for the relationship between them. In particular, for continuous t-norms, we give a necessary and sufficient condition such that the type-I degree of symmetry and the type-II degree of symmetry are equal. And then, we obtain that the type-II degree of symmetry is more appropriate to character the degree of the symmetry of fuzzy relations than the type-I degree of symmetry with respect to whether they preserve the fuzzy e-equalities or not. In the meantime, for a special case with continuous t-norms, we provide a necessary and sufficient condition such that the type-I degree of symmetry preserves the fuzzy equalities. Finally, for conjunctive left-continuous idempotent uninorms with neutral element e(0,1], we find out a symmetric fuzzy relation S which is close enough to the given fuzzy relation R with respect to fuzzy e-equality such that the degree of fuzzy e-equality of S and R is the type-II degree of symmetry of R. In particular, we obtain analogous results for continuous t-norms and one class of left-continuous t-norms.

Section snippets

A brief review on the development of the indicators of fuzzy relations

The notion of fuzzy relation [48] was introduced by Zadeh in 1965. A fuzzy relation R on a set X is a fuzzy subset of X×X, i.e., a mapping R:X×X[0,1]. It is one of the most fundamental concepts in fuzzy set theory and is widely investigated in the theory and applications of fuzzy mathematics, see, e.g., [6], [36], [38], [50]. For instance, the reflexivity, symmetry, &-transitivity, etc. are intensely investigated, see, e.g., [6], [21], [37], [49].

For a fuzzy relation, it is difficult to

Preliminaries

In this section, we give some basic concepts related to uninorms, t-norms and fuzzy relations, and some properties related to them, which will be used in the sequel.

The degree of the symmetry of fuzzy relations based on conjunctive left-continuous uninorms

In this section, first, we introduce two degrees of the symmetry of fuzzy relations based on conjunctive left-continuous uninorms. One is the type-I degree of symmetry as and the other is the type-II degree of symmetry ds. And then, we investigate the properties of those two degrees.

The approximations of non-symmetric fuzzy relations

In this section, we consider a special topic in the research of the degree of the symmetry of fuzzy relations, namely, the approximations of non-symmetric fuzzy relations. More precisely, we will find a symmetric fuzzy relation S which is close enough to the given fuzzy relation R with respect to fuzzy e-equality EX2 such that the degree of fuzzy e-equality of S and R is the type-II degree of symmetry ds of R. It is shown that for conjunctive left-continuous idempotent uninorms with neutral

Concluding remarks

In this paper, we investigate the degree of the symmetry of fuzzy relations from the mathematical point of view. The main results of this paper are listed as follows.

  • For any conjunctive left-continuous uninorm U with neutral element e(0,1], based on the fuzzy e-equality EX2 on the set consisting of fuzzy relations, we propose the two different degree of the symmetry of fuzzy relations. One is the type-I degree of symmetry as and the other is the type-II degree of symmetry ds. Note that, if U

Acknowledgements

The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant Nos. 11571010 and 61179038).

References (50)

  • H. Lai et al.

    Fuzzy preorder and fuzzy topology

    Fuzzy Sets Syst.

    (2006)
  • H. Lai et al.

    Closedness of the category of liminf complete fuzzy orders

    Fuzzy Sets Syst.

    (2016)
  • Y.M. Li et al.

    Remarks on uninorm aggregation operators

    Fuzzy Sets Syst.

    (2000)
  • S.V. Ovchinnikov

    Similarity relations, fuzzy partitions, and fuzzy ordering

    Fuzzy Sets Syst.

    (1991)
  • L. Valverde

    On the structure of F-indistinguishability operators

    Fuzzy Sets Syst.

    (1985)
  • X. Wang et al.

    Indicators of fuzzy relations

    Fuzzy Sets Syst.

    (2013)
  • X. Wang et al.

    Traces and property indicators of fuzzy relations

    Fuzzy Sets Syst.

    (2014)
  • R. Yager et al.

    Uninorm aggregation operators

    Fuzzy Sets Syst.

    (1996)
  • L.A. Zadeh

    Fuzzy sets

    Inf. Control

    (1965)
  • L.A. Zadeh

    Similarity relations and fuzzy orderings

    Inf. Sci.

    (1971)
  • M. Baczyński et al.

    Fuzzy Implications, Studies in Fuzziness and Soft Computing, vol. 231

    (2008)
  • L. Běhounek

    Extensionality in graded properties of fuzzy relations

  • R. Bělohlávek

    Fuzzy Relational Systems, Foundations and Principles

    (2002)
  • U. Bodenhofer

    Fuzzy orderings of fuzzy sets

  • D. Boixader et al.

    Fuzzy Equivalence Relation: Advanced Material

    (2000)
  • Cited by (0)

    View full text