Meticulous rough inclusion and its relations to fuzzy inclusion

We discuss a special kind of rough inclusions and its relation to fuzzy inclusions. One of four conditions of rough inclusions is detailed and a rough inclusion which satisfies the detailed condition as well as an additional condition is called a 'meticulous rough inclusion'. We show that a meticulous rough inclusion is closed with respect to upper and lower closure operators defined for a rough inclusion. We discuss conditions for a rough inclusion to be a meticulous rough inclusion. Moreover, the relations between meticulous rough inclusions and fuzzy inclusions are investigated. Sufficient conditions for a fuzzy inclusion defined by an implication function to be a meticulous rough inclusion are given. Examples of fuzzy inclusions which are meticulous rough inclusions are presented.