Subordination operator

Symbol-free definition
The subordination operator is a map from the subgroup property space to itself that sends a subgroup property $$p$$ to the property of being a subgroup for which there exists a ascending chain of subgroups from the subgroup to the group with each member satisfying $$p$$ in its successor.

Definition with symbols
The subordination operator on a property $$p$$ gives the following property: $$H$$ satisfies it in $$G$$ if there is an ascending chain $$H = H_0$$ &le; $$H_1$$ &le; $$... H_n = G$$ with each $$H_i$$ satisfying $$p$$ in $$H_{i+1}$$.

Transitive and identity-true
As for a general Kleene star operator, the subordination operator is a monotone descendant operator and is also idempotent. The fixed points are precisely the transitive identity-true properties.