The necessary and sufficient conditions for a fuzzy relation being ⊤-Euclidean

The connections between special types of fuzzy relations and properties of fuzzy rough approximation operators have been established in recent years, but ⊤-Euclidean fuzzy relation has not been considered yet. In the present paper, the necessary and sufficient conditions for a fuzzy relation being ⊤-Euclidean are investigated in three different fuzzy rough approximation spaces. Meanwhile, it is proved that in (I, ⊤)-fuzzy rough approximation space, where I is an R-implication, the properties the ⊤-Euclidean (I, ⊤)-fuzzy rough approximation operators possess are just the same as those in rough fuzzy approximation space.