Generators of aggregation functions and fuzzy connectives

Abstract

We show that the class of all aggregation functions on $[0,1]$ can be generated as a composition of infinitary sup-operation $\bigvee$ acting on sets with cardinality not exceeding $\mathfrak{c}$, $b$-medians $\mathsf{Med}_b$, $b\in[0,1[$, and unary aggregation functions $1_{]0,1]}$ and $1_{[a,1]}$, $a\in ]0,1]$. 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.