Self-similar abelian groups and their centralizers

We extend results on transitive self-similar abelian subgroups of the group automorphisms $\mathcal{A}\_m$ an $m$-ary tree $\mathcal{T}\_m$ by Brunner and Sidki to general case where permutation induced first level tree, has $s\geq 1$ orbits. prove that such a $A$ embeds in $A^$ which is also maximal subgroup $\mathcal{A}\_m$. The construction $A^{}$ based definition free monoid $\Delta$ rank $s$ partial diagonal monomorphisms $\mathcal{A}m$. Precisely, $A^{} = \overline{\Delta(B(A))}$, $B(A)$ denotes product projections its action different orbits subtrees $\mathcal{T}\_m$, bar topological closure. Furthermore, we if non-trivial, then C{\mathcal{A}m} (\Delta(A))$, centralizer $\Delta(A)$ When torsion group, it shown necessarily finite exponent. Moreover, recent constructions groups infinite enumerable examples are $\Delta$-invariant for $s=2$. In final section, introduce $m=ns \geq 2$, generalized adding machine $a$, automorphism $\mathcal{T}{m}$, show $\mathcal{A}{m}$ be split extension $\langle \rangle^{}$ $\mathcal{A}\_s$. describe important $\mathbb{Z}\_n \[\mathcal{A}\_s]$ submodules a\rangle^{}$.