Remark

If $u(a_1, \ldots, a_{i-1}, 0, a_{i+1}, \ldots, a_n) = 0$ for $i = 1, 2, \ldots, n$ and $a_j \in E_j$ ($j \neq i$), then formula (2) remains true for families $(x_i, \lambda)_{\lambda \in L_i}$ of finite support.

A special case of Definition 5 is that where $u$ is the law of action associated with the action of a set $\Omega$ on a magma $E$. If $u$ is distributive with respect to the second variable, it is also said that the action of $\Omega$ on the magma $E$ is distributive. In other words: