## Stream: maths

### Topic: complete (distrib) lattice

#### Kenny Lau (Jun 23 2019 at 22:29):

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

#### 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

thanks

#### Kenny Lau (Jun 23 2019 at 22:32):

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

#### 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?

#### Kenny Lau (Jun 23 2019 at 22:39):

yeah I want it to be a morphism of complete lattice

#### Kevin Buzzard (Jun 23 2019 at 22:40):

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

#### 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.

#### 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.

#### 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

#### Kevin Buzzard (Jun 23 2019 at 22:45):

So that's just a one way implication

#### Kevin Buzzard (Jun 24 2019 at 07:38):

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

#### Kevin Buzzard (Jun 24 2019 at 07:39):

I don't really know what an inf is, but the assertion that $p\le x\sqcap y$ iff $p\le x$ and $p\le y$ 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]

\cap

#### Mario Carneiro (Jun 24 2019 at 07:41):

the actual notation in lattice theory textbooks is \and

#### Mario Carneiro (Jun 24 2019 at 07:41):

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

#### 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

#### Kevin Buzzard (Jun 24 2019 at 07:44):

\cap

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

\sqcap

#### 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.

#### 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

#### Mario Carneiro (Jun 24 2019 at 07:52):

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

#### Mario Carneiro (Jun 24 2019 at 07:53):

I always forget which way slices go

#### 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