Right residual operator for composition

Property-theoretic statement
Suppose $$p,q$$ are two subgroup properties. The right residual of $$p$$ by $$q$$ is the unique subgroup property $$r$$ such that:

$$q * s \le p \iff s \le r$$.

Here, $$*$$ denotes the composition operator.

Statement with symbols
Suppose $$p,q$$ are two subgroup properties. The right residual of $$p$$ by $$q$$ is defined as the subgroup property $$r$$ as follows:

$$H$$ has property $$r$$ in $$G$$ if whenever $$K$$ is a subgroup of $$H$$ with property $$q$$ in $$H$$, $$K$$ has property $$p$$ in $$G$$.

Relation with right transiter
The right transiter of a subgroup property $$p$$ is defined as the right residual of $$p$$ by itself.