The algebraic structures known as bags were introduced by R. Yager as set-like algebraic structures where elements are allowed to be repeated. Since the original papers by Yager, different definitions of the concept of fuzzy bag, and the corresponding operators, are available in the literature, as well as some extensions of the union, intersection and difference operators of sets, and new algebraic operators.In general, the current definitions of bag pose very interesting issues related to the ontological aspects and practical use of bags. In this paper, we introduce a characterization of bags viewing them as the result of a count operation on the basis of a mathematical correspondence. We also discuss on the extension of our alternative characterization of bags to the fuzzy case. On these basis, we introduce some operators on bags and fuzzy bags, and we compare them to existing approaches. Finally, we deal with the case where no information about the correspondence is available, and only bounds can be provided for the count of elements of the result of algebraic operators. For this purpose, the notion of IC-bag (Chakrabarty, in: Proc.