Zulip Chat Archive

Stream: maths

Topic: complete (distrib) lattice


view this post on Zulip Kenny Lau (Jun 23 2019 at 22:29):

Is there a complete lattice that is not a complete distrib lattice?

view this post on Zulip Reid Barton (Jun 23 2019 at 22:30):

I think the standard example is a lattice with a bottom, top, and 3 incomparable elements between them

view this post on Zulip Kenny Lau (Jun 23 2019 at 22:32):

thanks

view this post on Zulip Kenny Lau (Jun 23 2019 at 22:32):

can every complete distrib lattice be embedded into the powerset of a set?

view this post on Zulip Kevin Buzzard (Jun 23 2019 at 22:35):

It wouldn't surprise me if every partial order could be embedded into the power set of a set. But do you want things like infs to coincide or something?

view this post on Zulip Kenny Lau (Jun 23 2019 at 22:39):

yeah I want it to be a morphism of complete lattice

view this post on Zulip Kevin Buzzard (Jun 23 2019 at 22:40):

Are you sure there's not just some dumb construction which gives it?

view this post on Zulip Kevin Buzzard (Jun 23 2019 at 22:42):

I don't really know what lattices look like. What is the definition of a morphism of complete lattices? Because there are relations involved rather than functions I never know whether a morphism is an if or an iff on the relations.

view this post on Zulip Kevin Buzzard (Jun 23 2019 at 22:43):

It's not like group theory where you know f(gh)=f(g)f(h) must be right.

view this post on Zulip Reid Barton (Jun 23 2019 at 22:43):

Usually the conditions you consider are "preserves infs" or "preserves sups" or both. Besides preserving the ordering itself obviously

view this post on Zulip Kevin Buzzard (Jun 23 2019 at 22:45):

So that's just a one way implication

view this post on Zulip Kevin Buzzard (Jun 24 2019 at 07:38):

How about sending xx in the poset PP to the subset {pPpx}\{p\in P\,|\,p\le x\} of PP?

view this post on Zulip Kevin Buzzard (Jun 24 2019 at 07:39):

I don't really know what an inf is, but the assertion that pxyp\le x\sqcap y iff pxp\le x and pyp\le y is the assertion that the intersection works. [I don't even know how to do glb in LaTeX, that's how un-lattice I am] [edit: fixed]

view this post on Zulip Mario Carneiro (Jun 24 2019 at 07:40):

\cap

view this post on Zulip Mario Carneiro (Jun 24 2019 at 07:41):

the actual notation in lattice theory textbooks is \and

view this post on Zulip Mario Carneiro (Jun 24 2019 at 07:41):

we sort of co-opted the "square meet/join" operators for use in lean

view this post on Zulip Mario Carneiro (Jun 24 2019 at 07:43):

To answer your question, morphisms in order categories are generally only one direction, e.g. monotone functions

view this post on Zulip Kevin Buzzard (Jun 24 2019 at 07:44):

\cap

Oh I know how to do intersections! I meant the weird square one.

view this post on Zulip Mario Carneiro (Jun 24 2019 at 07:44):

\sqcap

view this post on Zulip Kevin Buzzard (Jun 24 2019 at 07:48):

Now I don't know how to proceed because I don't know the axioms for sqcap :-) Seems like a sensible approach though. The construction might not even be injective for preorders but it is for partial orders.

view this post on Zulip Kevin Buzzard (Jun 24 2019 at 07:50):

It's sort of baby Yoneda. You get a predicate on your category from an object x by sending y to the prop of homs from y to x

view this post on Zulip Mario Carneiro (Jun 24 2019 at 07:52):

I guess it's a slice category in category speak?

view this post on Zulip Mario Carneiro (Jun 24 2019 at 07:53):

I always forget which way slices go

view this post on Zulip Alec Edgington (Jun 26 2019 at 08:16):

Hello,

can every complete distrib lattice be embedded into the powerset of a set?

Apparently you don't even need completeness for this to be true (preserving meets and joins), but you do need the axiom of choice (or a weak form of it).


Last updated: May 06 2021 at 17:38 UTC