Different generalizations of bags

We consider four different generalizations of bags (alias multisets). We first discuss Yager’s fuzzy bags having different sets of operations. It is shown that one is not a generalization of fuzzy sets but a mapping of them into fuzzy bags, since operations are inconsistent between the two, while the other includes fuzzy sets as particular cases. Third type is called real-valued bags which is simpler than the former two and is a kind of the reduction of fuzzy bags. Finally, the fourth generalization called G-bags includes all three except the first type. It is a minimal extension of the second and the third generalizations. Bag relations are defined for the third type of real-valued bags, which can further be generalized for G-bags.