An algebraic classification of means

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