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 (''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''⁺, 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''⁺.}}

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''⁺) 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''⁺ is said to be '''unperforated''' if ''n'' · ''g'' ∈ ''G''⁺ for some positive integer ''n'' implies ''g'' ∈ ''G''⁺. Being unperforated means there is no "gap" in the positive cone ''G''⁺.

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 l: ℓ-group).

A '''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''₁, ''x''₂, ''y''₁, ''y''₂ are elements of ''G'' and ''x_i'' ≤ ''y_j'', then there exists ''z'' ∈ ''G'' such that ''x_i'' ≤ ''z'' ≤ ''y_j''.

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.

Partially ordered groups are used in the definition of valuations of fields.

== Examples ==

* The integers 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 '''Z'''^''n'', where the group operation is componentwise addition, and we write (''a''₁,...,''a''_''n'') ≤ (''b''₁,...,''b''_''n'') if and only if ''a''_''i'' ≤ ''b''_''i'' (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 K₀(''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''^''n'' ≤ ''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, 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 group is already abelian. This has to do with the fact that a directed group is embeddable into a 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 |author-link=Garrett Birkhoff |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|url-access=subscription }} *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|last1=Blyth|first1=T.S.| 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