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I’ve been adding some things I’ve recently learned to prime ideal theorem.
I’ve just finished doing a massive rewrite of prime ideal theorem which does things in pretty fair (but certainly not ultimate) generality, mostly from the point of view of quantale theory (and so I’ve added a little more to that article, too). I think I’m done for the time being. Enough details are there so that anyone could follow.
Sample theorem from prime ideal theorem: if $T$ is a commutative Lawvere theory, then every proper ideal in a monoid of $Set^T$ is contained in a prime ideal. This gives for example the PIT for rigs, which had been on a sort of to-do list in an earlier version of the article.
And I decided to do a little reorganization at ideal under “Kinds of ideals”, adding a little more material along the way.
Wow, very cool! I don’t have time to read it all carefully, but this looks very nice. How much of this did you come up with yourself?
Thanks very much, Mike! Except for the bit at the end of prime ideal theorem giving some general categorical conditions for when to expect a PIT (and which I thought might be a little bit ad hoc when I mentioned it recently in connection with ideals in a monoid), the rest I believe has been known for quite a while (Paseka’s article – once I deciphered it – was particularly helpful recently). However, much of the literature that I’ve scanned on this seems not too friendly to the outsider, and it took some effort for me to put this together. A lot of reading between the lines, as it were.
And yet, it’s a fairly simple story I’ve tried to tell (the arrangement of which is where I’ll take credit), although there are still some aspects I’d like to understand better, naturally.
It’s a really nice story to have on the nLab all in one place and clearly explained.
On a barely related note, somehow it’s puzzling to me that a compact locale must have a point. I don’t intuitively see the connection between open-cover compactness and consistency of a theory.
I sympathize with that puzzlement. As noted in the article, the result is also in Stone Spaces (I.9), but I haven’t combed through Johnstone’s development to see whether the intuition behind it shines through.
Oh, duh, is it that any nontrivial locale represents a finitely consistent theory, so if it is compact then it must be globally consistent as well? (There must be some nontriviality condition in the result, since the empty space is compact.)
That’s probably a good way to think of it, although that hadn’t occurred to me!
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