Flex-norms: aggregation of fuzzy arguments

The rules of a fuzzy system work as follows: as the input variable x increases, the result of the application of this rule at first increases, reaches a maximum and then decreases (as a consequence of the triangular, Gaussian, etc. shapes of fuzzy sets). To describe this phenomenon, a new class of norms is introduced: flex-norms, which describe the aggregation of fuzzy sets, where the arguments are the underlying variables, and not degrees of membership. It is shown that these norms are commutative and have an identity e which lies in the interval [0,1]. This class of norms is similar to the class of uni-norms (monotonic, associative, commutative, and having identity in the interval [0,1]). Based on this similarity, a general formula and the main properties describing the class of flex-norms is established.