digit vectors.

Every nonnegative integer $k$, $0\le k < 2^m$, has a unique representation in base $2$ of the form $k=k_0+k_1 2 + \dots +k_{m-1} 2^{m-1}$ with digits $k_j\in \mathcal{A}_2$, $0\le j\le m-1$.

We identify $\mathcal{A}_2^m$ with the group ${\ensuremath{\mathbb{Z}}}/2^m{\ensuremath{\mathbb{Z}}}$ as follows. For ${\ensuremath{\mathbf x}}\in \mathcal{A}_2^m$, ${\ensuremath{\mathbf x}}= (x_0, \ldots, x_{m-1})$ , we define the map ${\ensuremath{\mathrm{int}}}_2: \mathcal{A}_2^m \rightarrow {\ensuremath{\mathbb{Z}}}/2^m {\ensuremath{\mathbb{Z}}}$, $${\ensuremath{\mathrm{int}}}_2({\ensuremath{\mathbf x}}) = x_0+ x_1 2+\dots + x_{m-1}2^{m-1}.$$ Further, let $\operatorname{dig}_2: {\ensuremath{\mathbb{Z}}}/2^m {\ensuremath{\mathbb{Z}}}\rightarrow \mathcal{A}_2^m$, $$\operatorname{dig}_2(k) = (k_0,  k_1, \ldots , k_{m-1}),$$ where $k=k_0 + k_1 2 + \dots + k_{m-1}2^{m-1}$ is the representation of $k$ in base 2. Finally, for ${\ensuremath{\mathbf x}}, {\ensuremath{\mathbf y}}\in \mathcal{A}_2^m$, we define $${\ensuremath{\mathbf x}}+ {\ensuremath{\mathbf y}}= \operatorname{dig}_2 \left( {\ensuremath{\mathrm{int}}}_2({\ensuremath{\mathbf x}}) + {\ensuremath{\mathrm{int}}}_2({\ensuremath{\mathbf y}}) \pmod{2^m}\right).$$

With this binary operation, the pair $(\mathcal{A}_2^m, +)$ is an abelian group. Clearly, it is isomorphic to the additive group ${\ensuremath{\mathbb{Z}}}/2^m {\ensuremath{\mathbb{Z}}}$. We call this type of binary operation [*addition with carry*]{} or [*integer addition*]{} of digit vectors.

For $m\ge 2$, our two examples of addition act on non-isomorphic groups, because ${\ensuremath{\mathbb{Z}}}/2^m{\ensuremath{\mathbb{Z}}}$ is cyclic and $({\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})^m$ is not. Apart from these two examples, are there any other possibilities to define addition on the set $\mathcal{A}_2^m$? From the Fundamental Theorem for 