Zulip Chat Archive

Stream: maths

Topic: Atoms in order and algebra


view this post on Zulip Aaron Anderson (Dec 20 2020 at 19:00):

I'd like to define the predicateis_atom in an order_bot, and is_coatom in an order_top, where is_atom means that an element is not bot but has nothing else below it, and conversely for is_coatom.

view this post on Zulip Aaron Anderson (Dec 20 2020 at 19:01):

These happen also to be the definitions of minimal and maximal submodules/ideals.

view this post on Zulip Aaron Anderson (Dec 20 2020 at 19:02):

I'd then like to rephrase ideal.is_maximal to be in terms of is_coatom. I'm guessing the correct way to do that would be with an abbreviation, so we keep the term ideal.is_maximal, or at least submodule.is_maximal?

view this post on Zulip Aaron Anderson (Dec 20 2020 at 19:04):

Also, I'd like to define a predicate (probably a class) on bounded_lattices, indicating that there are only two elements. This would be equivalent to is_atom top or is_coatom bot. Then this could be used to describe simple modules and several other kind of "simple" objects in algebra.

view this post on Zulip Aaron Anderson (Dec 20 2020 at 19:04):

What would be a good name for this predicate?

view this post on Zulip Kevin Buzzard (Dec 20 2020 at 19:07):

is_simple? That's what it's called in group theory.

view this post on Zulip Aaron Anderson (Dec 20 2020 at 19:09):

I guess one approach might even be to define it as having exactly two distinct elements... in which case it'd live near things like nontrivial and unique in the library

view this post on Zulip Aaron Anderson (Dec 20 2020 at 19:09):

but I'm not sure how useful that is elsewhere, so I'd be fine with just defining is_simple on bounded lattices, or really something simultaneously extending has_bot and has_top.


Last updated: May 11 2021 at 16:22 UTC