Some results on the degree of symmetry of fuzzy relations
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 , i.e., a mapping . 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 and the other is the type-II degree of symmetry . 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 such that the degree of fuzzy e-equality of S and R is the type-II degree of symmetry 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.
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For any conjunctive left-continuous uninorm U with neutral element , based on the fuzzy e-equality 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 and the other is the type-II degree of symmetry . 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).
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