Definition talk:Stable under Intersection

Isn't this just a specific instance of Definition:Closure (Abstract Algebra)? That is, do we need a separate page for it? We have already established that $\left({\mathbb S, \cap}\right)$ (where $\mathbb S \subseteq \mathcal P \left({S}\right)$ can be considered as an algebraic structure, so I contend that this definition is probably superfluous. --prime mover 05:49, 23 March 2012 (EDT)
Well, it is, but then where would we legitimately define the notation $\cap$-stable, which is a convenient shorthand? And do we want to bash innocent readers with abstract algebra when they are dealing with sets and intersections only? A reference to that page, i.e. 'That is, $\left({\mathcal S, \cap}\right)$ is a magma.' would be legitimate. --Lord_Farin 06:17, 23 March 2012 (EDT)