Definition talk:Stable under Intersection

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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)
You see, I instantly think of the concept as "closed under intersection". Naively, I would have assumed that someone doing measure theory would already be familiar with the basics of abstract algebra, but then perhaps not. A link to magma sounds good, and it's also probably worth putting an "also see" in closure to here.
Interestingly, it's the same concept as for a topological space. The same concept is introduced throughout the whole of mathematics, but each time it's treated afresh, as something new and isolated. What you lose out on is the ability to take on board a whole raft of results appropriate to the general monoid, which could bring insights that might otherwise not be apparent. Methinks there's a grand Kleinian unification task that's needed to be done ... --prime mover 09:40, 23 March 2012 (EDT)
The problem with building up mathematics is that one needs considerable experience before one's mind permits one abstraction after the other. Probably, that is why every field on some points starts all over. I have been in a bit of a dichotomy as to whether consistently 'stable' or 'closed' should be used; in any case, both are viable, but the saying 'a system of sets is stable' makes more intuitive sense than '.. is closed'. But of course, the two are equivalent and we should try and use/mention these facts as much as possible. Otherwise, abstract algebra just stands a bit on itself without the far-reaching implications it actually has being attributed to it. Same goes for category theory, but we haven't really ventured there yet. All in all, I get a to-be-suppressed urge to go through all of PWs results and definitions and link them to abstract algebra; however, I haven't got half a year to spare... Maybe it's good to first start rewriting, completing and amending Help:Editing cs. to make more people capable of embarking on this apparently endless journey towards rigour, consistency and readability; that's probably sort of what you mean by a Kleinian unification task. --Lord_Farin 09:50, 23 March 2012 (EDT)