# 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: *a* ≤ *b* if and only if -*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*, then there exists

_{j}*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

[edit]- 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 K_{0}(*A*) is a partially ordered abelian group. (Elliott, 1976)

## Properties

[edit]### Archimedean

[edit]The Archimedean property of the real numbers can be generalized to partially ordered groups.

- Property: A partially ordered group is called
**Archimedean**when for any , if and for all then . Equivalently, when , then for any , there is some such that .

### Integrally closed

[edit]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.^{[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.^{[2]}
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.^{[1]}

## See also

[edit]- Cyclically ordered group – Group with a cyclic order respected by the group operation
- Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb
- Ordered field – Algebraic object with an ordered structure
- Ordered ring – ring with a compatible total order
- Ordered topological vector space
- Ordered vector space – Vector space with a partial order
- Partially ordered ring – Ring with a compatible partial order
- Partially ordered space – Partially ordered topological space

## Note

[edit]## References

[edit]- M. Anderson and T. Feil,
*Lattice Ordered Groups: an Introduction*, D. Reidel, 1988. - Birkhoff, Garrett (1942). "Lattice-Ordered Groups".
*The Annals of Mathematics*.**43**(2): 313. doi:10.2307/1968871. 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. - Glass, A. M. W. (1982).
*Ordered Permutation Groups*. doi:10.1017/CBO9780511721243. ISBN 9780521241908. - Glass, A. M. W. (1999).
*Partially Ordered Groups*. ISBN 981449609X. - 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. - Kopytov, V. M.; Medvedev, N. Ya. (1994).
*The Theory of Lattice-Ordered Groups*. doi:10.1007/978-94-015-8304-6. 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. *Lattices and Ordered Algebraic Structures*. Universitext. 2005. doi:10.1007/b139095. ISBN 1-85233-905-5., chap. 9.- Elliott, George A. (1976). "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras".
*Journal of Algebra*.**38**: 29–44. doi:10.1016/0021-8693(76)90242-8.

## Further reading

[edit]Everett, C. J.; Ulam, S. (1945). "On Ordered Groups". *Transactions of the American Mathematical Society*. **57** (2): 208–216. doi:10.2307/1990202. JSTOR 1990202.

## External links

[edit]- Kopytov, V.M. (2001) [1994], "Partially ordered group",
*Encyclopedia of Mathematics*, EMS Press - Kopytov, V.M. (2001) [1994], "Lattice-ordered group",
*Encyclopedia of Mathematics*, EMS Press *This article incorporates material from partially ordered group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*