Generators of Aggregation Functions and Fuzzy Connectives

We show that the class of all aggregation functions on [0, 1] can be generated as a composition of infinitary sup-operation V acting on sets with cardinality not exceeding c, b-medians Med_b, b ∈ [0, 1[, and unary aggregation functions 1_{]0, 1]} and 1_{[a, 1]}, a ∈ ]0, 1]. Moreover, we show that we cannot relax the cardinality of argument sets for suprema to be countable, thus showing a kind of minimality of the introduced generating set. As a by-product, generating sets for fuzzy connectives, such as fuzzy unions, fuzzy intersections, and fuzzy implications, are obtained, too.