Sims-reduced generating set

Definition
Let $$G$$ be a group acting on a set $$S = \{ 1, 2, 3, \ldots, n \}$$ of size $$n$$ (equivalent $$G$$ is equipped with an embedding in the symmetric group on $$n$$ elements). A Sims-reduced generating set is a generating set for $$G$$ where for any elements $$i < j$$, there is at most one elements of $$G$$ which fixes all elements less than $$i$$ and sends $$i$$ to $$j$$.

Any generating set for $$G$$ can be trimmed down to a Sims-reduced generating set using the Sims filter.

Stronger properties

 * Jerrum-reduced generating set
 * Strong generating set