The set of fuzzy relative integers and fuzzy bags

Abstract

A characterization of fuzzy bags with conjunctive fuzzy integers (N f ) provides a general framework in which sets, bags, fuzzy sets, and fuzzy bags are treated in a uniform way. In this context, the difference between two fuzzy bags does not always exist and, in such cases, only approximations of the difference have been defined. The problem comes from the fact that the fuzzy bag model considered so far is based on positive fuzzy integers. In this paper, we construct the set of fuzzy relative integers (Z f ) where the existence of negative multiplicities allows us to define exact complementations. Then we define fuzzy bags on Z f , which offer a well-founded framework in which the difference of two fuzzy bags is always defined. C 2009 Wiley Periodicals, Inc.