Editing Partially ordered group (section)

==Properties==

=== Archimedean ===
The Archimedean property of the real numbers can be generalized to partially ordered groups.

:Property: A partially ordered group <math>G</math> is called '''Archimedean''' when for any <math>a, b \in G</math>, if <math>e \le a \le b</math> and <math>a^n \le b</math>  for all <math>n \ge 1</math> then <math>a=e</math>. Equivalently, when <math>a \neq e</math>, then for any <math>b \in G</math>, there is some <math>n\in \mathbb{Z}</math> such that <math>b < a^n</math>.

=== Integrally closed ===
A partially ordered group ''G'' is called '''integrally closed''' if for all elements ''a'' and ''b'' of ''G'', if ''a''<sup>''n''</sup> ≤ ''b'' for all natural ''n'' then ''a'' ≤ 1.<ref name=Glass>{{harvtxt|Glass|1999}}
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This property is somewhat stronger than the fact that a partially ordered group is [[Archimedean property|Archimedean]], though for a [[lattice-ordered group]] to be integrally closed and to be Archimedean is equivalent.<ref>{{harvtxt|Birkhoff|1942}}</ref>
There is a theorem that every integrally closed [[directed set|directed]] group is already [[abelian group|abelian]].  This has to do with the fact that a directed group is embeddable into a [[complete lattice|complete]] lattice-ordered group if and only if it is integrally closed.<ref name=Glass/>