A version of Calder\'{o}n-Mityagin theorem for the class of rearrangement invariant groups

Let $l_0$ be the group (with respect to the coordinate-wise addition) of all sequences of real numbers $x=(x_k{)}_{k=1}^{\mathrm{\infty }}$ that are eventually zero, equipped with the quasi-norm $\mathrm{\parallel }x{\mathrm{\parallel }}_0=\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}\{suppx\}$. A description of orbits of elements in the pair $(l_0,l_1)$ is given, which complements (in the sequence space setting) the classical Calder'{o}n-Mityagin theorem on a description of orbits of elements in the pair $(l_1,l_{\mathrm{\infty }})$. As a consequence, we obtain that the pair $(l_0,l_1)$ is $\mathcal{K}$-monotone.