An algebraic classification of means

Given a real interval \(I\), a group of homeomorphisms \(\mathcal{G} \left(M,I\right)\) is associated to every continuous mean defined \(i\)n \(I\). Two means \(M\), \(N\) defined in \(I\) will belong to the same class when \(\mathcal{G} (M, I) = \mathcal{G} (N,I)\). The equivalence relation defined in this way in \(\mathcal{CM}(I)\), the family of continuous means defined in \(I\), gives a principle of classification based on the algebrai object \(\mathcal{G}(M, I)\). Two major questions are raised by this classification: 1) the problem of computing \(\mathcal{G} (M, I)\) for a given mean \(M \in \mathcal{CM} (I)\), and 2) the determination of general properties of the means belonging to a same class. Some instances of these questions will find suitable responses in the present paper.