Example

(1) Let E be an associative magma written multiplicatively. The mapping which associates with a strictly positive integer $n$ the mapping $x \mapsto x^n$ of E into itself is an action of $\mathbf{N}^*$ on E. If E is a group, the mapping which associates with a rational integer $a$ the mapping $x \mapsto x^a$ of E into E is an action of $\mathbf{Z}$ on E.

(2) Let E be a magma with law denoted by $\top$. The mapping which associates with $x \in E$ the mapping $A \mapsto x \top A$ of the set of subsets of E into itself is an action of E on $\mathscr{P}(E)$.

(3) Let E be a set. The identity mapping of $E^E$ is an action of $E^E$ on E, called the canonical action. The corresponding law of action is the mapping $(f, x) \mapsto f(x)$ of $E^E \times E$ into E.

(4) Let $(\Omega_i)_{i \in I}$ be a family of sets. For all $i \in I$, let $f_i : \Omega_i \to E^E$ be an action of $\Omega_i$ on E. Let $\Omega$ be the sum of the $\Omega_i$ (Set Theory, II, § 4, no. 8). The mapping $f$ of $\Omega$ onto $E^E$, extending the $f_i$, is an action of $\Omega$ on E. This allows us to reduce the study of a family of actions to that of a single action.

(5) Given an action of $\Omega$ on E with law denoted by $\perp$, a subset $\Xi$ of $\Omega$ and a subset X of E, $\Xi \perp X$ denotes the set of $\alpha \perp x$ with $\alpha \in \Xi$ and $x \in X$; when $\Xi$ consists of a single element $\alpha$, we generally write $\alpha \perp X$ instead of $\{\alpha\} \perp X$. The mapping which associates with $\alpha \in \Omega$ the mapping $X \mapsto \alpha \perp X$ is an action of $\Omega$ on $\mathscr{P}(E)$, which is said to be derived from the given action by extension to the set of subsets.

(6) Let $\alpha \mapsto f_\alpha$ be an action of $\Omega$ on E. Let g be a mapping of $\Omega'$ into $\Omega$. Then the mapping $\beta \mapsto f_{g(\beta)}$ is an action of $\Omega'$ on E.

(7) Let $f : E \times E \to E$ be a law of composition on a set E. The mapping $\gamma : x \mapsto \gamma_x$ (resp. $\delta : x \mapsto \delta_x$) (§ 2, no. 2) which associates with the element $x \in E$ left (resp. right) translation by $x$ is an action of E on itself; it is called the left (resp. right) action of E on itself derived from the given law. When $f$ is commutative, these two actions coincide.

The law of left (resp. right) action associated with $\gamma$ is $f$ (resp. the opposite law to $f$). The law of right (resp. left) action associated with $\delta$ is $f$ (resp. the opposite law to $f$).

Let $\Omega, E, F$ be sets, $\alpha \mapsto f_\alpha$ an action of $\Omega$ on E and $\alpha \mapsto g_\alpha$ an action of $\Omega$ on F. An $\Omega$-*morphism of E into F*, or mapping of E into F compatible with the action of $\Omega$, is a mapping $h$ of E into F such that

for all $\alpha \in \Omega$ and $x \in E$. The composition of two $\Omega$-morphisms is an $\Omega$-morphism.

Let $\Omega, \Xi, E, F$ be sets, $\alpha \mapsto f_\alpha$ an action of $\Omega$ on $E$, $\beta \mapsto g_\beta$ an action of $\Xi$ on $F$ and $\phi$ a mapping of $\Omega$ into $\Omega'$. A $\phi$-morphism of $E$ into $F$ is a mapping $h$ of $E$ into $F$ such that

for all $\alpha \in \Omega$ and $x \in E$.