Talk:Upper set

Proposed merge

Please discuss this on the talk page of lower set. — Tobias Bergemann 08:51, 28 August 2006 (UTC)

OK, I did the merge. The two articles had slightly different wording for their examples, so for now I left them both. This will be confusing, since the upper and lower sets are really the same, but someone with more math theory than me should figure out which wording is the best, and use it for both definitions.Lisamh 21:01, 19 September 2006 (UTC)

Do you reckon we should change the name of the article to 'Upper sets and lower sets' or similar? Joel Brennan (talk) 16:35, 19 April 2019 (UTC)

Why partially ordered set?

The article says "... an upper set is a subset Y of a given partially ordered set (X,≤) such that, ...". But the condition that the set is partially ordered is too strong. A preorder is sufficient for the possibility that a upper set exists. —The preceding unsigned comment was added by 217.225.228.95 (talk) 19:19, 31 December 2006 (UTC).

You're right. We could add a section to tell that. But in practice it has very little use, since by defining the equivalence relation A≈B iff A≤B and B≤A and taking the quotient space of X under this relation (X/≈), once gets a partial order ←. You can then find the upper sets of (X,≤) from the upper sets of (X,←). Moreover, some properties and definitions of upper sets are easier to define/demonstrate, or only valid in the context of partial orders.--Clément Pillias (talk) 20:06, 6 November 2008 (UTC)

Isomorphic to abstract simplicial complex?

Am I right that upper sets of a partially ordered set Y with |Y|=n are isomorphic to the abstract simplicial complexes over n vertices? If so, it seems reasonable to add cross-references. Hv (talk) 11:39, 10 August 2010 (UTC)

Principal lower sets?

The article currently says, "A lower set is called principal if it is of the form ↓{x} where x is an element of X." Is this really supposed to say lower set? It makes almost no sense in terms of motivation/usefulness. For the case of power sets, a principal lower set is then just the specified singleton and the empty set. Who cares enough about this structure to single it out? Perhaps I have misunderstood something.? It makes much more sense for upper sets, and indeed ultrafilters (which are upper sets) are called principal just in case they have this form (some cluster point x). Also, are upper sets of the form ↑{x} not called principal? Cheers, Honestrosewater (talk) 04:05, 10 January 2012 (UTC)

I think you're missing a level of set containership. The principal lower set defined by a singleton set {x} of a power set P(X) would be ↓{{x}}, not ↓{x}. The principal lower set defined by any other set S in in a power set P(X) is ↓{S}. S does not have to be a singleton here. In the article, it writes "↓{x}" but in that case x ias an element of a partially ordered set, not an element of the universe of a powerset. —David Eppstein (talk) 04:47, 10 January 2012 (UTC)

Oh, right, right. I misread that. Cheers, Honestrosewater (talk) 05:29, 10 January 2012 (UTC)

Upper set properties references?

The article state : "Conversely any antichain A determines an upper set {x: for some y in A, x ≥ y}. For partial orders satisfying the descending chain condition this correspondence between antichains and upper sets is 1-1, but for more general partial orders this is not true". Can you give a reference for this property ? — Preceding unsigned comment added by 78.198.27.23 (talk) 14:30, 18 October 2017 (UTC)

Sorry for my ignorance, and sorry if I reply in the wrong place, but as I read this books description of Upper Set,

https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf

then isn’t the Upper Sets of the Powerset of {1,2,3,4}, the sets {1,2,3,4}, {2,3,4}, {3,4} and {4}?

I mean, how can {1} be an Upper Set in itself, since according to the definition then all elements larger than 1 is included in the Upper Set? 94.18.238.170 (talk) 16:50, 9 December 2022 (UTC)

The power set is the family of all subsets. It is partially ordered by set inclusion. The numerical values are irrelevant to its definition. —David Eppstein (talk) 19:39, 9 December 2022 (UTC)

Thanks for responding on this and sorry, I think I failed to express my question clearly.

I was refering to the hasse diagram exampple on this page, ordering the power set of {1, 2, 3, 4} and marking the Upper Sets in the diagram in green.

Here e.g. {1} has been marked in green to represent an Upper Set as I read the text.

The question is that I fail to understand how e.g. {1} can be an Upper Set given the definition and an Upper Set?

My understanding from reading the definition and the referenced book, is that the following sets should have been marked in green (i.e. as Upper Sets) {1,2,3,4}, {2,3,4}, {3,4} and {4}?

But maybe I am misunderstanding the purpose of the coloring? 94.18.238.170 (talk) 04:45, 10 December 2022 (UTC)

The diagram does not show {1} being an upper set. It shows {{1},{1,2},{1,3},{1,4},{1,2,3},{1,2,4},{1,3,4},{1,2,3,4}} as being an upper set. Upper sets are defined for things that have a partial order on them. The "things" here are themselves sets, like {1}, and the partial order on them is set inclusion, like {1} ⊆ {1,2,4}. It is an upper set because whenever a thing like {1} is in it, the bigger things like {1,2,4} are also in it. —David Eppstein (talk) 05:42, 10 December 2022 (UTC)

Beautiful explanation. Thanks a lot! I by mistake thought that the preorder relation (I hope I use the wording correctly) were on the natural numbers. But now I see, as you wrote in your previous comment, that the preorder relation is on the set inclusion. I hope I got it. Again, thanks a lot for the clarification! 94.18.238.170 (talk) 08:06, 10 December 2022 (UTC)

And the diagram is showing the upward closure of the element {1}. I hope I got it. 94.18.238.170 (talk) 08:23, 10 December 2022 (UTC)

I suggest to change to a relation on natural numbers ("is divisor of"), rather than on sets, to avoid the set-of-sets confusion above. - Jochen Burghardt (talk) 15:47, 10 December 2022 (UTC)

Yes, I think that was unnecessarily confusing. —David Eppstein (talk) 18:34, 10 December 2022 (UTC)