On relationship between interval type-2 fuzzy rough sets and interval type-2 fuzzy topologies induced by interval type-2 fuzzy relations

This paper investigates the relationship between interval type-2 fuzzy rough sets and Alexandrov interval type-2 fuzzy topologies induced by special interval type-2 fuzzy relations. We first show that every serial interval type-2 fuzzy relation generates an Alexandrov interval type-2 fuzzy topology. It is then observed that the lower and upper approximation operators induced by merely an interval type-2 fuzzy reflexive relation do not necessarily constitute interior and closure operators, respectively. To overcome this issue, we consider the notion of transitive closure of interval type-2 fuzzy relation and prove that every reflexive interval type-2 fuzzy relation and its transitive closure induce identical topologies. This provides a new approach for constructing interval type-2 fuzzy topologies from relations that are not necessarily transitive. Furthermore, for similarity (reflexive and symmetric) interval type-2 fuzzy relations, we introduce the concept of equivalence closure and establish that a similarity relation and its equivalence closure generate the same topology. Finally, we characterize the relationship between interior and closure operators associated with the resulting Alexandrov interval type-2 fuzzy topologies and interval type-2 fuzzy relations.