Generalized bags and their relations: An alternative model for fuzzy set theory and applications

Bags alias multisets have been known to be a fundamental tool for information system models. Hence bags have been studied for a long time by famous computer scientists. 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 complements, s-norms of bags, and bag relations. In the first part, we consider real-valued bags. After briefly reviewing basic relations and operations of classical bags, we introduce two types of complementation operations, and then introduce s-norms and t-norms of bags. A key idea is to use the infinite point into the domain of membership values. Fundamental properties such as duality of s-norms and t-norms are shown. As a result, an s-norm of a Minkowski type and its dual t-norm are derived. Another useful tool is bag relations. We define three types of compositions of max-s, max-t, and min-s operations for bag relations and prove that the compositions can be handled like matrix calculations. We moreover mention applications of bag relations to networks and data analysis, and suggest possible applications of bags to decision making using convex functions. In the second part, we study a class of generalized bags that are smallest extension of real-valued bags and fuzzy bags. It is proved that the generalized bags are in a sense equivalent to fuzzy number-valued bags. Using alpha-cuts, many operations of real-valued bags except a complementation are generalized to the corresponding operations of generalized bags, and fundamental properties are proved.