Topological structures of L-fuzzy rough sets and similarity sets of L-fuzzy relations

This paper investigates the topological structures of L-fuzzy rough sets. In particular, the family of lower (resp. upper) sets with respect to an arbitrary L-fuzzy relation is proved to be an Alexandrov L-fuzzy topology. Lower and upper similarity sets of L-fuzzy relations are studied with the transitive closure of L-fuzzy relation to investigate L-fuzzy relations inducing the same Alexandrov L-fuzzy topology by lower and upper sets, respectively. Moreover, the transitive subsets of lower and upper similarity sets of the same L-fuzzy relation coincide with each other, which are shown to be complete distributive lattices.