Partially-defined equivalence relations: Relationship with orthopartitions and connection to rough sets

We introduce partially -defined equivalence relations as a type of equivalence relation that incorporates uncertainty. In these relations, certain pairs of objects are not definitively determined to be related or unrelated. The relationship with orthopartitions is put forward, providing the conditions under which an orthopartition can be transformed into an equivalent partially -defined equivalence relation and vice versa. Additionally, we explore their connection with reducts in rough set theory, offering insights into the characterization of similarity reducts in terms of orthopartitions.