Generalized multiset theory.

The paper gives a formal (axiomatic) theory of collections, such that elements may occur some number of times in these collections, and this number is allowed to be an arbitrary real number; e.g., a negative one. The presented theory is a generalization of multiset theories and some fuzzy set theories. Some important consequences of the theory are revealed. Among potential applications of the theory, expressing of vague properties and relations is anticipated. The usefulness of the theory and its consequences for applications involving vague properties is demonstrated by an example of multi-criteria decision making based on vague properties of alternatives.