Partially 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: if and only if

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

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

:Property: A partially ordered group ''G'' is called Archimedean when ''a''^''n'' ≤ ''b'' for all natural ''n'' then ''a'' = ''e''. Equivalently, when ''a''≠''e'', then for any ''b''∈''G'', there is some $n\in \mathbb$ such that ''b'' <''a''^''n''.

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.

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.
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.

See also

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Note

References

*M. Anderson and T. Feil, ''Lattice Ordered Groups: an Introduction'', D. Reidel, 1988.
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*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.
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*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.
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*R. B. Mura and A. Rhemtulla, ''Orderable groups'', Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
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Further reading

External links

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*{{PlanetMath attribution
, urlname=PartiallyOrderedGroup , title=partially ordered group

Ordered algebraic structures
Ordered groups
Order theory