Operations for Real-Valued Bags and Bag Relations

A generalization of bags is proposed and fundamental operations are described. That is, a real-valued bag is proposed which is a straightforward generalization of the traditional bags, but is necessary for complementation, and s-norm and t-norm operations. We mainly describe how complementation and s-norms are related. As a result a duality theorem between s-norm and t-norm is obtained. R-bag relations with max-s norm composition and maxt norm composition are also proposed. Further generalization that includes fuzzy bags and real-valued bags having membership of a region in a plane is moreover discussed. Keywords— bag relation, max-s composition, real-valued bag, set-valued bag, t-norm and conorm .