A graded notion of functionality

We will introduce the notion of functionality as a graded property of a fuzzy relation. We will extend known results relating to functional fuzzy relations to their graded counterparts and provide new properties w.r.t. various kinds of relational compositions. As a contribution to the theory of fuzzy control, we will moreover provide an example of use of graded properties in establishing implicative models of fuzzy rule bases where the non-graded property fails.