Generators of aggregation functions and fuzzy connectives

TL;DR

This paper demonstrates that all aggregation functions on [0,1] can be constructed from a specific set of basic operations, establishing a minimal generating framework and deriving fuzzy connectives as a consequence.

Contribution

It introduces a minimal generating set for all aggregation functions on [0,1], including infinitary sup, b-medians, and unary functions, and shows the minimality of argument set cardinality.

Findings

All aggregation functions can be generated from the specified basic operations.

Countability of argument sets for suprema cannot be relaxed, proving minimality.

Fuzzy connectives like unions, intersections, and implications are derived as a byproduct.

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.