Definition 2

A subset $A$ of a set $E$ is called stable under an action $\alpha \mapsto f_\alpha$ of $\Omega$ on $E$ if $f_\alpha(A) \subset A$ for all $\alpha \in \Omega$. An element $x$ of $E$ is called invariant under an element $\alpha$ of $\Omega$ if $f_\alpha(x) = x$.

The intersection of a family of stable subsets of $E$ under a given action is stable. There therefore exists a smallest stable subset of $E$ containing a given subset $X$ of $E$; it is said to be generated by $X$; it consists of the elements $(f_{\alpha_1} \circ f_{\alpha_2} \circ \cdots \circ f_{\alpha_n})(x)$, where $x \in X, n \geq 0, \alpha_i \in \Omega$ for all $i$.