Talk:Semilattice

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Please forgive me for asking a question that I should already know the answer to. I am puzzling over the following statement in the article on the Semilattice:

"Using an easy induction argument, one can also conclude the existence of all suprema of non-empty finite subsets in any join-semilattice."

Couldn't this be rephrased as: "By induction, one can identify all suprema in any non-empty finite subset within any join-semilattice."

I realize this is a stronger statement, but if it cannot be made then what would be a sound but weaker version? I am particularly having problems with the phrase: "... conclude the existence of all ...". This brings to mind some composition of existential and universal quantifiers that I need to somehow know about.

I appreciate any help I can get on this. Thanks Vonkje 15:59, 26 May 2005 (UTC)

The phrase is meant as follows: If I know that a poset has all binary suprema, then I also know that it has all finite suprema (that are not empty). This is shown by re-writing a finite supremum like V{x1, x2, x3, ..., xn} to a chain of binary suprema (...(x1 v x2) v x3) ...) v xn). To do this formally, we have to apply induction. This in turn is easy, once you have the idea.

I will try to simplify the statement in the article a little. --Markus Krötzsch 21:06, 31 August 2005 (UTC)

PS: Isn't most of mathematics about asking questions that we should already know the answer to? --Markus Krötzsch 21:06, 31 August 2005 (UTC)

[edit] Something odd

Hi,

As it appears you state x^0 = x ... and a bit latter "use 1 for v".

Looking at the "boolean algebra" page, one of course will find the more traditional x^1 = x, and xv0=x.

Wouldn't it be nice to change this to be conformant to the boolean algebra page.

Or may be I missed something (I am not a mathematician, rather a nerd).

I share your confusion. I believe the passage you are refering to reads:

To define meet-semilattices with greatest elements, one introduces an additional constant 0, that makes (S,,0) an idempotent, commutative monoid. Explicitly, one requires that

x 0 = x

for all x in S, in addition to (S, ) being a meet-semilattice as introduced before. Again, join-semilattices with least element are defined similarly, although in this case one prefers 1 to denote the neutral element.

I think it should be amended to read:

To define meet-semilattices with least elements, one introduces an additional constant 1, that makes (S,,1) an idempotent, commutative monoid. Explicitly, one requires that

x 1 = x

for all x in S, in addition to (S, ) being a meet-semilattice as introduced before. Again, join-semilattices with greatest element are defined similarly, although in this case one prefers 0 to denote the neutral element.

The bolded words above only denote changes and will not be bolded if the article were to be amended. I have also dropped a note at the guy from Dresden's page to take a look at this. Thanks for calling this to our attention.

--Vonkje 21:58, 10 Jun 2005 (UTC)

Sorry for the late reply. I need emails rather than talk pages to respond quickly ;-) Of course both of you were right: if numbers are used at all, greatest elements are certainly denoted as "1" and least elements as "0". I guess I do not have to explain why. It was just a typo. Be bold in updating pages (though mathematicals are sometimes treacherous and it might be better to discuss with someone in the Mathematics community). Anyway, thanks again for the note on my talk page. --Markus Krötzsch 21:06, 31 August 2005 (UTC)

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Is a semilattice medial? If not, is there a special name for a medial semilattice?--SurrealWarrior 21:11, 22 January 2006 (UTC)

[edit] Could someone supply a reference or two?

I have polished the syntax of the entire entry except the section Free Semilattices While I am no algebraist, I do own a copy of Davey and Priestley (2003). I invite readers with more mathematical training than I have to correct any mathematical errors and confusions, either remaining or that I have inadvertantly slipped in.

Semilattices fascinate me because I see them as the bridge linking Algebra I and II: semilattices are groupoids (not in the category theory sense!) with idempotence. This entry has no proper references. As far as I can determine, semilattices are very poorly covered in the usual intro monographs and there are no specialized monographs.202.36.179.65 18:28, 24 April 2006 (UTC)

Surely PT Johnstone's classic textbook Stone Spaces should be supplied as a reference? It contains a rigorous and thorough treatment of semilattices. —The preceding unsigned comment was added by 90.192.192.3 (talk) 14:30, 25 February 2007 (UTC).

[edit] Binary operations

I added Semilattice to the category Binary operations, but on viewing the list of these I got cold feet. In general, this community seems to have separated the operation in se from the structure (set,operation). Thus, I think it is more correct to add pages Meet (mathematics) and Join (mathematics), and add them to the binary operations category.

On the other hand, the meet and join pages should refer back to semilattices...

A check of a preview shows that the suggested pages exist - but only as redirects to Lattice (order). I do think writing separate pages is better - not only to be able to enumerate meet and join as binary operations, but also since they appear not only in lattices but also in semilattices. JoergenB 14:22, 20 September 2006 (UTC)