Definition:Saturation (Group Action)

Definition

Let $G$ be a group acting on a set $X$.

Let $S\subset X$ be a subset.

Definition 1

The saturation of $S$ is its saturation by the equivalence relation induced by the action.

Definition 2

The saturation of $S$ is the union of its images under the group action:

$\overline S = \ds \bigcup_{g \mathop \in G} g S$

Also denoted as

The saturation is also denoted $\operatorname{Sat} \left({S}\right)$.

Also see

• Equivalence of Definitions of Saturation Under Group Action
• Definition:Saturation (Equivalence Relation)