Some results on the degree of symmetry of fuzzy relations

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.