Definition 1

Let $\Omega$ and E be two sets. A mapping of $\Omega$ into the set $E^E$ of mappings of E into itself is called an action of $\Omega$ on E.

Let $\alpha \mapsto f_\alpha$ be an action of $\Omega$ on E. The mapping $(\alpha, x) \mapsto f_\alpha(x)$ (resp. $(x, \alpha) \mapsto f_\alpha(x)$) is called the law of left (resp. right) action of $\Omega$ on E† associated with the given action of $\Omega$ on E. Given a mapping $g$ of $\Omega \times E$ (resp. $E \times \Omega$) into E, there exists one and only one action $\alpha \mapsto f_\alpha$ of $\Omega$ on E such that the associated law of left (resp. right) action is $g$ (Set Theory, II, § 5, no. 2, Proposition 3).

In this chapter we shall say, for the sake of abbreviation, "law of action"

† Or sometimes the external law of composition on E with $\Omega$ as operating set.

instead of "law of left action". The element $f_\alpha(x)$ of E (for $\alpha \in \Omega$ and $x \in E$) is sometimes called the transform of $x$ under $\alpha$ or the composition of $\alpha$ and $x$. It is often denoted by left (resp. right) multiplicative notation $\alpha . x$ (resp. $x . \alpha$), the dot may be omitted; the composition of $\alpha$ and $x$ is then called the product of $\alpha$ and $x$ (resp. $x$ and $\alpha$). The exponential notation $x^\alpha$ is also used. In the arguments of the following paragraphs we shall generally use the notation $\alpha \perp x$. The elements of $\Omega$ are often called operators.