Intersection-closed subgroup series property

Symbol-free definition
A subgroup series property is said to be intersection-closed if the intersection of any two subgroup series, both having the property, also has the property. Here, by intersection of subgroup series, we mean the member-wise intersection.

Definition with symbols
A subgroup series property $$p$$ is said to be intersection-closed if whenever $$S$$ and $$S'$$ are two subgroup series, each having property $$p$$ so does $$S \cap S'$$.

Related notions
A related notion to this is series-intersection-closed subgroup property, viz a subgroup property wherein the property of being a subgroup series wherein each member satisfies the property in its successor, is an intersection-closed subgroup series property.