The set of fuzzy rational numbers and flexible querying

A fuzzy bag is a bag in which each occurrence of an element is associated with a grade of membership. This notion can be viewed as a generalization of the concepts of set, fuzzy set and bag. The set of fuzzy integers (N-f) provides a general characterization in which all these different concepts are treated in a uniform way and can then be composed. In the field of databases, the use of fuzzy bags is motivated by their ability to manage both quantities and preferences. However, N-f becomes too restricted a framework when dealing with queries based on difference or division operations. So, a more general structure based on the set of fuzzy relative integers (Z(f)) in which exact differences can be performed, has been first developed. In this paper, we carry on with this approach and we extend Z(f) to the set of fuzzy rational numbers (Q(f)). This context leads to define a closed system of multiplicative operations and allows to perform exact divisions. Applied to flexible querying of databases, Q(f) and the notion of division on fuzzy numbers allow to generalize the relational division. They define a sound basis for queries involving ratios between quantities. (c) 2005 Elsevier B.V. All rights reserved.