[Merged by Bors] - feat(order/ideal, order/pfilter, order/prime_ideal): added `ideal_inter_nonempty`, proved that a maximal ideal is prime #6924

atarnoam · 2021-03-28T17:02:38Z

ideal_inter_nonempty: the intersection of any two ideals is nonempty. A preorder with joins and this property satisfies that its ideal poset is a lattice.
An ideal is prime iff x \inf y \in I implies x \in I or y \in I.
A maximal ideal in a distributive lattice is prime.

depends on: [Merged by Bors] - feat(order/ideal, order/pfilter, order/prime_ideal): added is_ideal, is_pfilter, is_prime, is_maximal, prime_pair #6656

While proving that a maximal ideal in a distributive lattice is prime, I've noticed a problem.
I wanted to use the sup of two ideals, however in the current implementation this requires P to have a bottom element!
What is going on here is that to prove that the sup ideal is nonempty, it suffices to use only the fact that the intersection of any two ideals is nonempty. This is true for both the case where P has bot, and for when P has infs.
Thus we can define this property separately and show that it is true for these two cases.

Co-authored-by: Bryan Gin-ge Chen <bryangingechen@gmail.com>

…rime_ideal

… into prime_ideal

…rime_ideal

… into prime_ideal

Co-authored-by: Mathieu Guay-Paquet <mathieu.guaypaquet@gmail.com>

To show that `ideal P` is a lattice, it now suffices to check that `P` has joins and that the intersection of any two ideals is nonempty. Also added an equivalent characteristic of the join ideal when `P` is a lattice.

… into prime_ideal

Co-authored-by: Bryan Gin-ge Chen <bryangingechen@gmail.com>

jcommelin

This is not really my area of expertise. So I'm just commenting on style.

src/order/ideal.lean

src/order/pfilter.lean

src/order/ideal.lean

Co-authored-by: Johan Commelin <johan@commelin.net>

mguaypaq

Sorry for the delay in reviewing, and thank you for this nice addition. I especially like your ideal_inter_nonempty, because I've wanted something like it before. As you noted, there are two natural and unrelated settings where this property holds (the two instances of it you define), and I don't think there's a better abstraction which covers both.

As always, I've tried to be somewhat thorough, so this review includes many comments, but most of them are minor changes due to the mathlib naming conventions and style guide.

src/order/ideal.lean

src/order/prime_ideal.lean

mguaypaq · 2021-04-09T18:33:53Z

src/order/prime_ideal.lean

+variables {I : ideal P}
+
+@[priority 100]
+instance is_maximal.is_prime [is_maximal I] : is_prime I :=


For such a nice theorem, and the main point of this PR, it's almost too bad that the statement is so unassuming!

mguaypaq

This PR seems fine to me, but a maintainer will have to poke bors so it can be merged.

awainverse · 2021-04-12T19:21:58Z

I think the ready-to-merge label indicates that a maintainer has already told bors to merge it, and that it's about to merge

bryangingechen · 2021-04-13T23:32:09Z

Looks like there's just one docstring suggestion left.

By the way, @atarnoam the way we've been doing reviews here, the author usually marks suggestions / conversations as "Resolved" if they've been addressed by commits or if the conversation has ended. This makes it easier for later reviewers to see at a glance what remains to be done.

@awainverse any final thoughts on this?

Co-authored-by: Mathieu Guay-Paquet <mathieu.guaypaquet@gmail.com>

bryangingechen · 2021-04-14T18:56:47Z

Thanks!
bors r+

…er_nonempty`, proved that a maximal ideal is prime (#6924) - `ideal_inter_nonempty`: the intersection of any two ideals is nonempty. A preorder with joins and this property satisfies that its ideal poset is a lattice. - An ideal is prime iff `x \inf y \in I` implies `x \in I` or `y \in I`. - A maximal ideal in a distributive lattice is prime.

bors · 2021-04-15T03:49:03Z

Pull request successfully merged into master.

Build succeeded:

atarnoam and others added 30 commits March 12, 2021 16:42

feat(order/ideal) added is_prime and is_maximal ideals

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starting to work on is_prime_of_is_maximal

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Update src/order/ideal.lean

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Co-authored-by: Mathieu Guay-Paquet <mathieu.guaypaquet@gmail.com>

Made the lattice instance of I more general.

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To show that `ideal P` is a lattice, it now suffices to check that `P` has joins and that the intersection of any two ideals is nonempty. Also added an equivalent characteristic of the join ideal when `P` is a lattice.

Changed proper to is_proper

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Merge branch 'prime_ideal' of github.com:leanprover-community/mathlib…

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is_prime, is_maximal now extend is_proper.

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made is_maximal and is_prime to typeclasses.

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Co-authored-by: Bryan Gin-ge Chen <bryangingechen@gmail.com>

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Co-authored-by: Bryan Gin-ge Chen <bryangingechen@gmail.com>

added is_ideal, is_filter, prime_pair

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Merge branch 'prime_ideal' into is_prime_of_is_maximal

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[Merged by Bors] - feat(order/ideal, order/pfilter, order/prime_ideal): added `ideal_inter_nonempty`, proved that a maximal ideal is prime #6924

[Merged by Bors] - feat(order/ideal, order/pfilter, order/prime_ideal): added `ideal_inter_nonempty`, proved that a maximal ideal is prime #6924

atarnoam commented Mar 28, 2021 •

edited by bryangingechen

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bors bot commented Apr 15, 2021

[Merged by Bors] - feat(order/ideal, order/pfilter, order/prime_ideal): added ideal_inter_nonempty, proved that a maximal ideal is prime #6924

[Merged by Bors] - feat(order/ideal, order/pfilter, order/prime_ideal): added ideal_inter_nonempty, proved that a maximal ideal is prime #6924

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bors bot commented Apr 15, 2021

[Merged by Bors] - feat(order/ideal, order/pfilter, order/prime_ideal): added `ideal_inter_nonempty`, proved that a maximal ideal is prime #6924

[Merged by Bors] - feat(order/ideal, order/pfilter, order/prime_ideal): added `ideal_inter_nonempty`, proved that a maximal ideal is prime #6924

atarnoam commented Mar 28, 2021 •

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