Talk:Partially ordered group

It seems easy to verify that if an ordered group is finite, the order has to be equality; i.e., apart from this trivial case an ordered group is infinite. If we have a source we should add this.--Patrick 07:52, 18 May 2007 (UTC)

Moreover, every element in the positive cone apart from the identity element has infinite order.--Patrick 08:04, 18 May 2007 (UTC)

+ for multiplication?

Why is + being used for multiplication here? As far as I know, orderability has nothing to do with abelian groups, so this notation is just...wrong.

Unless someone can come up with a good reason I will change this notation to be multiplication, as is standard.

Farpov (talk) 12:57, 5 August 2010 (UTC)

I would like to point out that the if and only if statement in the introduction of the article assumes G to be commutative as well. The correct statement for non-commutative G is: for all a,b in G: a<=b iff (-a*b is in G or b*-a is in G). Not assuming this results in the statement being false for all groups with non trivial centralizor!

I thus completely agree with Farpov. — Preceding unsigned comment added by 82.170.166.152 (talk) 20:29, 23 February 2012 (UTC)

In addition, things like "if a ∈ H then -x + a + x ∈ H for each x of G" become completely redundant for abelian groupsHagman (talk) 11:10, 11 December 2021 (UTC)

Abelian case

There's quite a lot of information behind partially ordered abelian groups that I would like to add. Should I add it to this page or create a new page? It may be distracting to have the abelian case here since the nonabelian case is quite different. Minimalrho (talk) 06:42, 7 January 2016 (UTC)

Upon further inspection of this article, I'm concerned about the extent to which the abelian and non-abelian cases are confused. Is perforation a useful concept in the non-abelian case? Do the definitions of Riesz groups and Riesz interpolation property require being abelian or is it standard terminology for non-abelian groups? Minimalrho (talk) 07:06, 12 January 2016 (UTC)

merge (Integrally closed) ?

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

Consensus to merge. Felix QW (talk) 07:02, 7 April 2022 (UTC)

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I suggest merging Integrally closed (Ordered group) into this article, because I think that Integrated closed will be explained in this article after explaining Archimedean. However, I think there is a way to write a short summary of Integrated closed in this article and not merge the articles.--SilverMatsu (talk) 07:13, 24 March 2022 (UTC)

Support the merger proposal; there is no need to spin out separate articles on individual properties of ordered groups as long as this articles is so bare of content, and readers interested in those properties benefit from viewing the material in the context of other stronger or weaker properties. In this case, it makes very good sense to keep Archmedean and Integrally closed together. Felix QW (talk) 10:02, 29 March 2022 (UTC)

@Felix QW: Thank you for your comment. We need to polish the article more, but the merge is complete anyway. --SilverMatsu (talk) 01:24, 7 April 2022 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Archimedean

Is there a source for this definition of archimedean? I checked the definitions in [0] and [1] and both require the stated property only for 1 ≤ a < b. This definition is not equivalent to the one from the article. In any case, the definition has to require at least that a be non-negative.

[0] V. M. Kopytov and N. Ya. Medvedev, Right-ordered groups, Siberian School of Algebra and Logic, Consultants Bureau, 1996.

[1] Glass, A. M. W. (1999). Partially ordered groups 2A00:1398:4:A0F:8877:8440:3C36:92D5 (talk) 09:41, 13 July 2023 (UTC)

Makes sense, i edited the definition to fit the references as you quote them. Thanks! jraimbau (talk) 06:52, 14 July 2023 (UTC)