Homomorphisms of the group \(L^1_s\) into the group \(L^1_r\) for \(r \leq 3\) (Q2474084)

Let \(s \in \mathbb N\). The group \(L^1_s\) is the set \[ Z_s=\{ \overline{x}_s=(x_1,\cdots,x_s) \in \mathbb R^s:\;x_1 \neq 0\;\} \] with the operation \(\overline{x}_s \overline{y}_s=\overline{z}_s\) defined by \[ z_n=\sum_{k=1}^n x_k \sum_{\overline{u}_n\in U_{n,k}} A_{\overline{u}_n} \prod_{j=1}^n y_j^{u_j}, \quad n=1,2,\cdots,s, \] where \[ U_{n,k}=\{ \overline{u}_n=(u_1,\cdots,u_n) \in \{0,1,\cdots,k\}^n:\;\sum_{j=1}^n u_j=k \quad \text{and}\quad \sum_{j=1}^n ju_j=n\;\} \] and \[ A_{\overline{u}_n}=\frac{n!}{\prod_{j=1}^n(u_j!(j!)^{u_j})}. \] The group \(L^1_s\) is called one-dimensional \(s\)th differential group. The homomorphisms of \(L^1_2\) into \(L^1_2\) and \(L^1_3\) have been determined in the paper ``On homomorphisms of the group \(L^1_2\) into the groups \(L^1_2\) and \(L^1_3\)'' by \textit{J. Chudziak}, \textit{P. Drygaś}, \textit{W. Jabłoński} and \textit{S. Midura} [Demonstr. Math. 31, No. 1, 143--152 (1998; Zbl 0952.39007)]. In the paper under review, the author determines the homomorphisms of \(L^1_s\) into \(L^1_1\), \(L^1_2\) and \(L^1_3\). In all cases the homomorphisms are built up by means of multiplicative and logarithmic functions.