On an Approximate-inclusion-based Difference Operator - Application to the Division of Fuzzy Relations

This paper is devoted to the definition of a difference operator founded on a relaxation of the inclusion. The regular inclusion of A in B entails that any element of A is also a member of B. The concept of an approximate inclusion is based on the softening of the universal quantifier (all) into "almost all". We show that such a weakened inclusion induces a drastic difference, whose main characteristics is to be non-deterministic. Despite this, it can be used in practice, which is illustrated in the area of databases, with the question of the algebraic rewriting of the (approximate) division of relations.