Nah. I'm thinking of it as "intersection" in the locale of Bousfield classes DL.

The question supposes that fundamental groups were defined before the unit interval...

I think the relevant paper here might be Hovey's paper on the chromatic splitting conjecture?

Ah yes Neil thankyou. I see the point now, where smash is a problem. I guess I'm trying to have some kind of descent property, so what you say may indeed work anyway. Thanks!

So Tyler, doesn't that orthogonality mean that we glue together an E(n) local spectrum really easily from it's K(n) local pieces?

And yes... you're right about the composition, to build the $E(n)$ localizations. I guess... hmm, what am I saying. I guess it should be something like that. In that case, it should be like wedging right? Since that's how we build our $E(n)$'s?

As well as situations with the Morava $K$ and $E$ theories.

I might add that there is the well known case where we do this with completion at primes and rationalization. I think...

Oh wow thanks so much! I had not heard of this book, but I will have to check it out.

Thankyou. I thus far have found Johnstone's book to be a little short on the basic equivalence of using filters versus using ideals. He seems to prefer ideals, but for my purposes filters are much better. I just wanted to make sure there was a good homeomorphism between Spec(A) and the topology on prime filters that you give so that I can basically use all the theorems for distributive lattices and Boolean algebras without thinking about whether or not I'm dealing with filters or ideals for the most part.