Some results on the degree of symmetry of fuzzy relations. (English) Zbl 1423.03207
Summary: 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 \(e \in(0, 1]\), 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.
MSC:
| 03E72 | Theory of fuzzy sets, etc. |
Keywords:
fuzzy relations; degree of the symmetry; fuzzy \(e\)-equalities; conjunctive left-continuous uninorms; left-continuous t-norms; \(RU\)-implicationsReferences:
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