On Two Approaches to Approximate Inclusion Operators

This paper is devoted to two extensions of the inclusion operator. The general idea advocated here is the tolerance for exceptions, thus defining an approximate inclusion. A first line for allowing for exceptions is quantitative and it is based on the weakening of the universal quantifier present in the logical view of the inclusion between regular or fuzzy sets. Another strategy is more qualitative and it is founded on the (graded) allowance for low-intensity exceptions. These two approaches are presented and discussed in terms of properties and their use for approximate divisions of relations serves as an illustration