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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJan 31st 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownContinuing through _[[Stone Spaces]]_, I'm adding material on free lattices from Section I.4. Just now, I've beefed up [[suplattice]] and started [[SemiLat]], but there is more to come.

   Continuing through Stone Spaces, I'm adding material on free lattices from Section I.4. Just now, I've beefed up suplattice and started SemiLat, but there is more to come.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeJan 31st 2010
   • (edited Feb 1st 2010)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownNow also [[SupLat]], [[Lat]], [[CompLat]], [[DistLat]], [[BoolAlg]], [[CompBoolAlg]], [[HeytAlg]], and (from Section II.1.2) [[Frm]].

   Now also SupLat, Lat, CompLat, DistLat, BoolAlg, CompBoolAlg, HeytAlg, and (from Section II.1.2) Frm.

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 1st 2010
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   Author: TobyBartels
   Format: MarkdownOK, that's it.

   OK, that's it.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 1st 2010
   • (edited Feb 1st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownThanks, Toby, for all this work! Now, it would be nice if somebody found the energy to collect all of these in a list at [[Stone Spaces]], too, maybe in a glossary section at the end...

   Thanks, Toby, for all this work!

   Now, it would be nice if somebody found the energy to collect all of these in a list at Stone Spaces, too, maybe in a glossary section at the end...

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 1st 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownThey're in the table of contents at [[Stone Spaces]], where Johnstone discusses them, in Section I.4. (I put them in the TOC as [[free lattice|free …]], since that is really what Johnstone is discussing about them. And other than the definitions, that is pretty much all that I say about them, although I named the articles after the categories in case people want to say more.)

   They're in the table of contents at Stone Spaces, where Johnstone discusses them, in Section I.4.

   (I put them in the TOC as free …, since that is really what Johnstone is discussing about them. And other than the definitions, that is pretty much all that I say about them, although I named the articles after the categories in case people want to say more.)

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 2nd 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownI just created [[Loc]] too. That is *very* stubby, but I'll be adding to it, I expect.

   I just created Loc too. That is very stubby, but I'll be adding to it, I expect.

7. • CommentRowNumber7.
   • CommentAuthorJohn Baez
   • CommentTimeMar 5th 2019
   • PermaLink
   Author: John Baez
   Format: MarkdownItexI have a bunch of related questions: 1) What's the free sup-semilattice on a poset? 2) What's the free complete sup-semilattice on a poset? 3) What's the free lattice on a poset? 4) What's the free complete lattice on a poset? 5) What's the free distributive lattice on a poset? 6) What's the free complete distributive lattice on a poset? I would hope that at least some of these have nice answers, maybe just in the finite case. I'm not having much luck finding them!

   I have a bunch of related questions:

   1) What’s the free sup-semilattice on a poset?

   2) What’s the free complete sup-semilattice on a poset?

   3) What’s the free lattice on a poset?

   4) What’s the free complete lattice on a poset?

   5) What’s the free distributive lattice on a poset?

   6) What’s the free complete distributive lattice on a poset?

   I would hope that at least some of these have nice answers, maybe just in the finite case. I’m not having much luck finding them!

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeMar 5th 2019
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   Author: Mike Shulman
   Format: MarkdownItexI think the free complete sup-semilattice (i.e. [[suplattice]]) on a poset $P$ consists of downsets in $P$. Probably the free sup-semilattice on $P$ likewise consists of downsets that are the union of finitely many principal ones. I don't know the others offhand. Have you looked at *Stone Spaces*? IIRC there's a fairly detailed study of lots of kinds of free posetal structures in there.

   I think the free complete sup-semilattice (i.e. suplattice) on a poset $P$ consists of downsets in $P$. Probably the free sup-semilattice on $P$ likewise consists of downsets that are the union of finitely many principal ones. I don’t know the others offhand. Have you looked at Stone Spaces? IIRC there’s a fairly detailed study of lots of kinds of free posetal structures in there.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMar 5th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI suppose mostly the answers there are for free structures on sets rather than posets (as recorded on the pages mentioned at the top of this thread), but probably they could be adapted.

   I suppose mostly the answers there are for free structures on sets rather than posets (as recorded on the pages mentioned at the top of this thread), but probably they could be adapted.

10. • CommentRowNumber10.
    • CommentAuthorJohn Baez
    • CommentTimeMar 6th 2019
    • PermaLink
    Author: John Baez
    Format: MarkdownItexThanks! I figured downsets should be good for something, and it makes sense that downsets throw in arbitrary sups; should I be worried that the downsets of a poset forms a _distributive_ lattice, or is this what we'd expect from the free suplattice in this case. It's well-known that the existence of the arbitrary sups in a poset implies the existence of arbitrary infs, but do we get distributivity 'for free' as well? Surely someone has carefully pondered the adjunctions between all our favorite categories of posets. If not, there must be fun left here! Maybe I should ask on the $n$-Café.

    Thanks! I figured downsets should be good for something, and it makes sense that downsets throw in arbitrary sups; should I be worried that the downsets of a poset forms a distributive lattice, or is this what we’d expect from the free suplattice in this case. It’s well-known that the existence of the arbitrary sups in a poset implies the existence of arbitrary infs, but do we get distributivity ’for free’ as well?

    Surely someone has carefully pondered the adjunctions between all our favorite categories of posets. If not, there must be fun left here! Maybe I should ask on the $n$-Café.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMar 6th 2019
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI'm pretty sure that a general suplattice is not automatically distributive. However it might still happen to be the case "by accident" that the *free* suplattice on a poset is always distributive.

    I’m pretty sure that a general suplattice is not automatically distributive. However it might still happen to be the case “by accident” that the free suplattice on a poset is always distributive.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 6th 2019
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexThe free suplattice on a poset $P$ is indeed the poset of downsets of $P$, and this is indeed distributive since it can be identified with the internal poset hom $[P^{op}, 2]$ which has meets and joins defined pointwise. Then, because $2$ is distributive, so must be $[P^{op}, 2]$. Another way of saying it: the set-theoretic intersection of two down-sets is a downset, and same for the set-theoretic union. Then use distributivity of set-theoretic operations. For the free distributive lattice on a poset $P$, I think it works like this. In the finite case, the functor $[-, 2]: FinPos^{op} \to FinDist$ is an equivalence. Thus the forgetful functor $U: FinDist \to Pos$ may be identified with $[-, 2]: FinPos^{op} \to FinPos$. The latter has a left adjoint $[-, 2]^{op}: FinPos \to FinPos^{op}$. Thus the left adjoint to $U$ becomes the composite $$FinPos \stackrel{[-, 2]^{op}}{\to} FinPos^{op} \stackrel{[-, 2]}{\to} FinDist$$ taking a poset $P$ to $[[P, 2], 2]$. This extends to a left adjoint $\Phi: Pos \to Dist$ by expressing an arbitrary poset as the filtered colimit of its finite subposets $F$: $$\Phi(P) = colim_{fin.\; F \hookrightarrow P} [[F, 2], 2].$$ As for the other questions: I'm not sure 2), 4), and 6) even exist in general, although I think you can get a variant of 6) if you use complete distributivity. I'm not sure what 3) looks like. It might actually be pretty complicated to describe (the free lattice on a set is a bit complicated, if I remember correctly).

    The free suplattice on a poset $P$ is indeed the poset of downsets of $P$, and this is indeed distributive since it can be identified with the internal poset hom $[P^{op}, 2]$ which has meets and joins defined pointwise. Then, because $2$ is distributive, so must be $[P^{op}, 2]$. Another way of saying it: the set-theoretic intersection of two down-sets is a downset, and same for the set-theoretic union. Then use distributivity of set-theoretic operations.

    For the free distributive lattice on a poset $P$, I think it works like this. In the finite case, the functor $[-, 2]: FinPos^{op} \to FinDist$ is an equivalence. Thus the forgetful functor $U: FinDist \to Pos$ may be identified with $[-, 2]: FinPos^{op} \to FinPos$. The latter has a left adjoint $[-, 2]^{op}: FinPos \to FinPos^{op}$. Thus the left adjoint to $U$ becomes the composite

    $$FinPos \stackrel{[-, 2]^{op}}{\to} FinPos^{op} \stackrel{[-, 2]}{\to} FinDist$$

    taking a poset $P$ to $[[P, 2], 2]$. This extends to a left adjoint $\Phi: Pos \to Dist$ by expressing an arbitrary poset as the filtered colimit of its finite subposets $F$:

    $$\Phi(P) = colim_{fin.\; F \hookrightarrow P} [[F, 2], 2].$$

    As for the other questions: I’m not sure 2), 4), and 6) even exist in general, although I think you can get a variant of 6) if you use complete distributivity.

    I’m not sure what 3) looks like. It might actually be pretty complicated to describe (the free lattice on a set is a bit complicated, if I remember correctly).

13. • CommentRowNumber13.
    • CommentAuthorJohn Baez
    • CommentTimeMar 7th 2019
    • PermaLink
    Author: John Baez
    Format: MarkdownItexThanks, Todd! Do you mind if I copy your reply over to the [$n$-Café discussion](https://golem.ph.utexas.edu/category/2019/03/left_adjoints_between_categori.html)? I think it may answer some questions I had over there about why some of the free constructions we're looking at seem to be 'squares' of certain functors, perhaps in about the same sense that your left adjoint $FinPos \to FinDist$ is the 'square' of $[-,2]$. ('Square' in inverted commas because of the tricky 'op' stuff.)

    Thanks, Todd! Do you mind if I copy your reply over to the $n$-Café discussion? I think it may answer some questions I had over there about why some of the free constructions we’re looking at seem to be ’squares’ of certain functors, perhaps in about the same sense that your left adjoint $FinPos \to FinDist$ is the ’square’ of $[-,2]$. (’Square’ in inverted commas because of the tricky ’op’ stuff.)

14. • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 7th 2019
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexHi John, Yes, that's fine. For some psychological reason I found it easier to write here than there, but anyway the discussions in both places are good and probably not too many people know this stuff.

    Hi John,

    Yes, that’s fine. For some psychological reason I found it easier to write here than there, but anyway the discussions in both places are good and probably not too many people know this stuff.

15. • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 7th 2019
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexIncidentally, 6) here is subtly different from 6) over at the Café, since here you say "complete" distributive lattice and there you say "complete*ly*" distributive lattice. So for 6) I would have answered differently over there.

    Incidentally, 6) here is subtly different from 6) over at the Café, since here you say “complete” distributive lattice and there you say “completely” distributive lattice. So for 6) I would have answered differently over there.

16. • CommentRowNumber16.
    • CommentAuthorJohn Baez
    • CommentTimeMar 8th 2019
    • (edited Mar 8th 2019)
    • PermaLink
    Author: John Baez
    Format: MarkdownItexOkay, thanks for the warning. I didn't know people deliberately used them to mean different things; I can imagine various concepts including: 1) all infs and sups exist, finite infs distribute over all sups and finite sups distribute over all infs 2) all infs and sups exist; all infs distribute over all sups and all sups distribute over all infs. Here I was using "complete distributive" to mean 2), aware of the potential ambiguity, but then I saw on Wikipedia they use "completely distributive" to mean 2) and that sounded better because it's not just distributive and complete. Now I'm even worrying that people could use "completely distributive" to mean "completely distributive but not necessarily complete", i.e. 3) finite infs and sups exist, all infs that exist distribute over all sups that exist and all sups that exist distribute over all infs that exist.

    Okay, thanks for the warning. I didn’t know people deliberately used them to mean different things; I can imagine various concepts including:

    1) all infs and sups exist, finite infs distribute over all sups and finite sups distribute over all infs

    2) all infs and sups exist; all infs distribute over all sups and all sups distribute over all infs.

    Here I was using “complete distributive” to mean 2), aware of the potential ambiguity, but then I saw on Wikipedia they use “completely distributive” to mean 2) and that sounded better because it’s not just distributive and complete. Now I’m even worrying that people could use “completely distributive” to mean “completely distributive but not necessarily complete”, i.e.

    3) finite infs and sups exist, all infs that exist distribute over all sups that exist and all sups that exist distribute over all infs that exist.

17. • CommentRowNumber17.
    • CommentAuthorJohn Baez
    • CommentTimeMar 8th 2019
    • (edited Mar 8th 2019)
    • PermaLink
    Author: John Baez
    Format: MarkdownItexBy the way, Todd, Richard Garner has answered all my questions, but I still like your answer because it shed some extra light on what's happening. Also by the way, the discussion on the n-Cafe of the nonexistent "free complete lattice on 3 generators" has made me really grok why the category of monads on Set doesn't have binary coproducts. Once that fact seemed surprising, but now it seems insane to hope for it without some "bounded rank" assumption. Of course our email conversations pushed me a long way down the road to understanding.

    By the way, Todd, Richard Garner has answered all my questions, but I still like your answer because it shed some extra light on what’s happening.

    Also by the way, the discussion on the n-Cafe of the nonexistent “free complete lattice on 3 generators” has made me really grok why the category of monads on Set doesn’t have binary coproducts. Once that fact seemed surprising, but now it seems insane to hope for it without some “bounded rank” assumption. Of course our email conversations pushed me a long way down the road to understanding.

18. • CommentRowNumber18.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 8th 2019
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItex> Now I’m even worrying Maybe some people do that, but I *think* most who contemplate these things follow the Tao and take "completely distributive" to include complete as well.

    Now I’m even worrying

    Maybe some people do that, but I think most who contemplate these things follow the Tao and take “completely distributive” to include complete as well.