Definition 6

An action $\alpha \mapsto f_\alpha$ of a set $\Omega$ on a magma $E$ is said to be distributive if, for all $\alpha \in \Omega$, the mapping $f_\alpha$ is an endomorphism of the magma $E$.

If $\top$ denotes the law of the magma $E$ and $\perp$ the law of action associated with the action of $\Omega$ on $E$, the distributivity of the latter is then expressed by the formula (6)

By an abuse of language, it is also said that the law $\perp$ is distributive (or right distributive) with respect to the law $\top$.

Formula (2) of § 1, no. 2 shows that then, for every ordered sequence $(x_\lambda)_{\lambda \in L}$ of elements of $E$ and every $\alpha \in \Omega$, (7)

If an action $\alpha \mapsto f_\alpha$ is distributive and an equivalence relation $R$ on $E$ is compatible with the law of composition of $E$ and the action $\alpha \mapsto f_\alpha$, the quotient action on $E/R$ is distributive.

When the law on $E$ is written multiplicatively, we often use the exponential notation $x^\alpha$ for a law of action which is distributive with respect to this multiplication, so that distributivity is expressed by the identity $(xy)^\alpha = x^\alpha y^\alpha$. If the law on E is written additively, we often use left (resp. right) multiplicative notation $\alpha . x$ (resp. $x . \alpha$) for a law of action which is distributive with respect to this addition, the distributivity being expressed by the identity

We may also consider the case where $\Omega$ has an internal law, denoted by $\overline{T}$, and the law of action is distributive with respect to the first variable, which means that

(8)

for all $\alpha, \beta \in \Omega$ and $x \in E$. Then, by formula (2) of § 1, no. 2

(9)

for every ordered sequence $(\alpha_{\lambda})_{\lambda \in L}$ of elements of $\Omega$ and all $x \in E$.