Example

Let E be a monoid (resp. group) written multiplicatively. The mapping $(n, x) \mapsto x^n$ of $\mathbf{N} \times E$ (resp. $\mathbf{Z} \times E$) into E is distributive with respect to the first variable by the equation $x^{m+n} = x^m x^n$ (with addition as law on $\mathbf{N}$). If E is commutative, this mapping is distributive with respect to the second variable by the equation $(xy)^n = x^n y^n$.