Definition 5

Let $E_1, \ldots, E_n$ and F be sets and u a mapping of $E_1 \times \cdots \times E_n$ into F. Let $i \in \{1, n\}$. Suppose $E_i$ and F are given the structures of magmas. u is said to be distributive relative to the index variable i if the partial mapping

is a homomorphism of $E_i$ into F for all fixed $a_j$ in $E_j$ and $j \neq i$.

If $\top$ denotes the internal laws on $E_i$ and F, the distributivity of u is given by the equations

(1)

for $i = 1, 2, \ldots, n, a_1 \in E_1, \ldots, a_{i-1} \in E_{i-1}, x_i \in E_i, x'_i \in E_i, a_{i+1} \in E_{i+1}, \ldots, a_n \in E_n$.