Graded Tolerant Inclusion and its Axiomatization

This paper is devoted to an extension of the inclusion operator. The main idea that we suggest is to relax the arguments of the inclusion by means of proximity-based modifiers in order to obtain a more tolerant inclusion. It is shown that the relaxation can be envisaged in order to design a tolerant inclusion whose result is either Boolean, or valued in the unit interval. The focus is particularly made on graded tolerant inclusion. A set of axioms that was previously proposed for the characterization of graded inclusion of fuzzy sets is revisited in order to take into account the specificities of tolerant inclusion.