Intersection-transiter

Symbol-free definition
The intersection transiter operator is a map from the subgroup property space to the subgroup property space, that acts as transiter for the commutative associative binary property operator namely the intersection operator. The intersection transiter takes a subgroup property $$p$$ to the property of being a subgroup whose intersection with any subgroup having property $$p$$, also has property $$p$$.

Definition with symbols
Let $$p$$ be a subgroup property. The intersection transiter operator of $$p$$ is defined as the following property $$q$$: a subgroup $$H$$ has property $$q$$ in $$G$$ if and only if for every subgroup $$K$$ with property $$p$$ in $$G$$, $$H \cap K$$ also has property $$p$$ in $$G$$.