Subsystems and automorphisms of some finite magmas of order $k+k^2$

This work is devoted to the study of subsystems of some finite magmas $\mathfrak{S}=(V,*) $ with a generating set of $k$ elements and order $k+k^2$. For $k>1$, the magmas $\mathfrak{S}$ are not semigroups and quasigroups. An element-by-element description of all magmas $\mathfrak{S}$ subsystems is given. It was found that all the magmas $\mathfrak{S}$ have subsystems that are semigroups. For $k>1$, subsystems that are idempotent nonunit semigroups are explicitly indicated. Previously, a description of an automorphism group was obtained for magmas $\mathfrak{S}$. In particular, every symmetric permutation group $S_k$ is isomorphic to the group of all automorphisms of a suitable magma $\mathfrak{S}$. In this paper, a general form of automorphism for a wider class of finite magmas of order $k+k^2 $ is obtained.