Generalized Bags and Bag Relations: Toward an Alternative Model for Fuzzy Set Applications

Bags which are also called multisets have been known to be a fundamental tool for information system models. Fuzzy bags have originally been proposed by Yager, and several researches about their applications have been done. Miyamoto established fundamental operations of fuzzy bags, and proposed generalized bags that include real-valued bags and fuzzy bags at the same time. Nevertheless, real usefulness of bag theory should be shown by studying bag relations. We show fundamental properties and operations of bags and bag relations. For the latter, we define three types of compositions and prove that the composition can be handled like matrix calculations. We moreover mention applications of bag relations to networks and data analysis, and suggest possible generalizations of fuzzy sets and rough approximations using bag theory.