On an idempotent transformation of aggregation functions and its application on absolutely continuous Archimedean copulas

In this paper, we isolate an interesting transformation of binary aggregation functions from a result on the propagation of transitivity in additive preference structures. This transformation is based on a projection technique and results in a smaller (or equal) binary aggregation function. The transformation is idempotent and acts internally on the classes of conjunctors, semi-copulas and quasi-copulas. Hence, also copulas are transformed into quasi-copulas, but in general not into copulas. Remarkably, the Frank copulas are mapped to new copulas.