Balanced fuzzy sets

• Published in: Information sciences 2006-09, Vol.176 (17), p.2467-2506
• Main Author: Homenda, Władysław
• Format: Article
• Language: English
• Subjects: Balanced norms; Fuzzy sets; Membership functions; Negative information; Triangular norms; Uninorms and nullnorms
• ISSN: 0020-0255; 1872-6291
• DOI: 10.1016/j.ins.2005.12.003

The paper presents a new approach to fuzzy sets and uncertain information based on an observation of asymmetry of classical fuzzy operators. Parallel is drawn between symmetry and negativity of uncertain information. The hypothesis is raised that classical theory of fuzzy sets concentrates the whole negative information in the value 0 of membership function, what makes fuzzy operators asymmetrical. This hypothesis could be seen as a contribution to a broad range discussion on unification of aggregating operators and uncertain information processing rather than an opposition to other approaches. The new approach “spreads” negative information from the point 0 into the interval [−1, 0] making scale and operators symmetrical. The balanced counterparts of classical operators are introduced. Relations between classical and balanced operators are discussed and then developed to the hierarchies of balanced operators of higher ranks. The relation between balanced norms, on one hand, and uninorms and nullnorms, on the other, are quite close: balanced norms are related to equivalence classes of some equivalence relation build on linear dependency in the spaces of uninorms and nullnorms. It is worth to stress that this similarity is raised by two entirely different approaches to generalization of fuzzy operators. This observation validates the generalized hierarchy of fuzzy operators to which both approaches converge. The discussion in this paper is aimed at presenting the idea and does not aspire to detailed exploration of all related aspects of uncertainty and information processing.