Definition of general aggregation operators through similarity relations

Various extensions of the original max and min aggregation operators in fuzzy set theory are successfully used in practical applications, but lack a clear conceptual model supporting them. Giving these operators a meaningful and simple interpretation is the topic of this paper. Aggregation operators are seen as different methods to measure distances to the essential reference points of the feature space, called Ideals. It has been proved that every general aggregation operator can be associated with a corresponding metric, in which the result of its application is the distance to the Ideal. Some widely used operators correspond to familiar l - p norms, and new operators can be defined by specifying different metrics. Heterogeneous combinations of ANDs and ORs are treated in such a way that the distributivity and De Morgan's laws hold. Applications to fuzzy constraint satisfaction problem and fuzzy control are discussed and interpreted geometrically. Classical operators are particular cases of the proposed semantic model, and several other examples are given.