 Solution

The additive identity for the shifted operation must satisfy $\mathbf{x}+0_{\oplus }-\mathbf{a}=\mathbf{x}$, giving $0_{\oplus }=\mathbf{a}$. The additive inverse must satisfy $\mathbf{x}+{\mathbf{x}}^{\prime }-\mathbf{a}=\mathbf{a}$, givingThe notation distinguishes the new additive identity from the ordinary zero vector. This is a translated abelian group structure on the same underlying set.