Editing Partially ordered group

{{Short description|Group with a compatible partial order}}
{{redirect|Ordered group|groups with a total or linear order|Linearly ordered group}}
In [[abstract algebra]], a '''partially ordered group''' is a [[group (mathematics)|group]] (''G'', +) equipped with a [[partial order]] "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' then ''a'' + ''g'' ≤ ''b'' + ''g'' and ''g'' +'' a'' ≤ ''g'' +'' b''.

An element ''x'' of ''G'' is called '''positive''' if 0 ≤ ''x''. The set of elements 0 ≤ ''x'' is often denoted with ''G''<sup>+</sup>, and is called the '''positive cone of ''G'''''.

By translation invariance, we have ''a'' ≤ ''b'' if and only if 0 ≤ -''a'' + ''b''.
So we can reduce the partial order to a monadic property: {{nobreak|''a'' ≤ ''b''}} [[if and only if]] {{nobreak|-''a'' + ''b'' ∈ ''G''<sup>+</sup>.}}

For the general group ''G'', the existence of a positive cone specifies an order on ''G''. A group ''G'' is a partially orderable group if and only if there exists a subset ''H'' (which is ''G''<sup>+</sup>) of ''G'' such that:
* 0 ∈ ''H''
* if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''a'' + ''b'' ∈ ''H''
* if ''a'' ∈ ''H'' then -''x'' + ''a'' + ''x'' ∈ ''H'' for each ''x'' of ''G''
* if ''a'' ∈ ''H'' and -''a'' ∈ ''H'' then ''a'' = 0

A partially ordered group ''G'' with positive cone ''G''<sup>+</sup> is said to be '''unperforated''' if ''n'' · ''g'' ∈ ''G''<sup>+</sup> for some positive integer ''n'' implies ''g'' ∈ ''G''<sup>+</sup>. Being unperforated means there is no "gap" in the positive cone ''G''<sup>+</sup>.

If the order on the group is a [[linear order]], then it is said to be a [[linearly ordered group]].
If the order on the group is a [[lattice order]], i.e. any two elements have a least upper bound, then it is a '''lattice-ordered group''' (shortly '''l-group''', though usually typeset with a [[Script typeface|script]] l: ℓ-group).

A '''[[Frigyes Riesz|Riesz]] group''' is an unperforated partially ordered group with a property slightly weaker than being a lattice-ordered group. Namely, a Riesz group satisfies the '''Riesz interpolation property''': if ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''y''<sub>1</sub>, ''y''<sub>2</sub> are elements of ''G'' and ''x<sub>i</sub>'' ≤ ''y<sub>j</sub>'', then there exists ''z'' ∈ ''G'' such that ''x<sub>i</sub>'' ≤ ''z'' ≤ ''y<sub>j</sub>''.

If ''G'' and ''H'' are two partially ordered groups, a map from ''G'' to ''H'' is a ''morphism of partially ordered groups'' if it is both a [[group homomorphism]] and a [[monotonic function]]. The partially ordered groups, together with this notion of morphism, form a [[category theory|category]].

Partially ordered groups are used in the definition of [[Valuation (algebra)|valuation]]s of [[field (mathematics)|field]]s.

== Examples ==

* The [[integer]]s with their usual order
* An [[ordered vector space]] is a partially ordered group
* A [[Riesz space]] is a lattice-ordered group
* A typical example of a partially ordered group is '''[[integer|Z]]'''<sup>''n''</sup>, where the group operation is componentwise addition, and we write (''a''<sub>1</sub>,...,''a''<sub>''n''</sub>) ≤ (''b''<sub>1</sub>,...,''b''<sub>''n''</sub>) [[if and only if]] ''a''<sub>''i''</sub> ≤ ''b''<sub>''i''</sub> (in the usual order of integers) for all ''i'' = 1,..., ''n''.
* More generally, if ''G'' is a partially ordered group and ''X'' is some set, then the set of all functions from ''X'' to ''G'' is again a partially ordered group: all operations are performed componentwise. Furthermore, every [[subgroup]] of ''G'' is a partially ordered group: it inherits the order from ''G''.
* If ''A'' is an [[approximately finite-dimensional C*-algebra]], or more generally, if ''A'' is a stably finite unital C*-algebra, then [[Approximately finite-dimensional C*-algebra#K0|K<sub>0</sub>]](''A'') is a partially ordered [[abelian group]]. (Elliott, 1976)

==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}}
</ref>

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/>

== See also ==

* {{annotated link|Cyclically ordered group}}
* {{annotated link|Linearly ordered group}}
* {{annotated link|Ordered field}}
* {{annotated link|Ordered ring}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Ordered vector space}}
* {{annotated link|Partially ordered ring}}
* {{annotated link|Partially ordered space}}

== Note ==
{{reflist}}

==References==

*M. Anderson and T. Feil, ''Lattice Ordered Groups: an Introduction'', D. Reidel, 1988.
*{{Cite journal |last=Birkhoff |first=Garrett |date=1942|title=Lattice-Ordered Groups |url=http://dx.doi.org/10.2307/1968871 |journal=The Annals of Mathematics |volume=43 |issue=2 |page=313 |doi=10.2307/1968871 |jstor=1968871 |issn=0003-486X}}
*M. R. Darnel, ''The Theory of Lattice-Ordered Groups'', Lecture Notes in Pure and Applied Mathematics 187, Marcel Dekker, 1995.
*L. Fuchs, ''Partially Ordered Algebraic Systems'', Pergamon Press, 1963.
*{{cite book |doi=10.1017/CBO9780511721243|title=Ordered Permutation Groups|year=1982|last1=Glass|first1=A. M. W.|isbn=9780521241908}}
*{{cite book |isbn=981449609X|title=Partially Ordered Groups|last1=Glass|first1=A. M. W.|year=1999|publisher=World Scientific |url={{Google books|5oTVCgAAQBAJ|Partially Ordered Groups|page=191|plainurl=yes}}}}
*V. M. Kopytov and A. I. Kokorin (trans. by D. Louvish), ''Fully Ordered Groups'', Halsted Press (John Wiley & Sons), 1974.
*V. M. Kopytov and N. Ya. Medvedev, ''Right-ordered groups'', Siberian School of Algebra and Logic, Consultants Bureau, 1996.
*{{cite book |doi=10.1007/978-94-015-8304-6|title=The Theory of Lattice-Ordered Groups|year=1994|last1=Kopytov|first1=V. M.|last2=Medvedev|first2=N. Ya.|isbn=978-90-481-4474-7}}
*R. B. Mura and A. Rhemtulla, ''Orderable groups'', Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
*{{cite book |doi=10.1007/b139095|title=Lattices and Ordered Algebraic Structures|series=Universitext|year=2005|isbn=1-85233-905-5}}, chap. 9.
*{{cite journal |doi=10.1016/0021-8693(76)90242-8|title=On the classification of inductive limits of sequences of semisimple finite-dimensional algebras|year=1976|last1=Elliott|first1=George A.|journal=Journal of Algebra|volume=38|pages=29–44|doi-access=}}

== Further reading ==
{{cite journal |doi=10.2307/1990202|jstor=1990202|title=On Ordered Groups|last1=Everett|first1=C. J.|last2=Ulam|first2=S.|journal=Transactions of the American Mathematical Society|year=1945|volume=57|issue=2|pages=208–216|doi-access=free}}

== External links ==
*{{Eom| title = Partially ordered group | author-last1 =Kopytov| author-first1 = V.M.| oldid = 48137}}
* {{Eom| title = Lattice-ordered group | author-last1 =Kopytov| author-first1 = V.M.| oldid = 47589}}
*{{PlanetMath attribution
 |urlname=PartiallyOrderedGroup |title=partially ordered group
}}

[[Category:Ordered algebraic structures]]
[[Category:Ordered groups]]
[[Category:Order theory]]