# 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 element** if 0 ≤ *x*. The set of elements 0 ≤ *x* is often denoted with *G*^{+}, and it is called the **positive cone of G**. So we have

*a*≤

*b*if and only if -

*a*+

*b*∈

*G*

^{+}.

By the definition, we can reduce the partial order to a monadic property: *a* ≤ *b* if and only if 0 ≤ -*a* + *b*.

For the general group *G*, the existence of a positive cone specifies an order on *G*. A group *G* is a partially ordered 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

- The integers
- 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)

## See also

## References

- M. Anderson and T. Feil,
*Lattice Ordered Groups: an Introduction*, D. Reidel, 1988. - 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. - A. M. W. Glass,
*Ordered Permutation Groups*, London Math. Soc. Lecture Notes Series 55, Cambridge U. Press, 1981. - 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. - V. M. Kopytov and N. Ya. Medvedev,
*The Theory of Lattice-Ordered Groups*, Mathematics and its Applications 307, Kluwer Academic Publishers, 1994. - R. B. Mura and A. Rhemtulla,
*Orderable groups*, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977. - T.S. Blyth,
*Lattices and Ordered Algebraic Structures*, Springer, 2005, ISBN 1-85233-905-5, chap. 9. - G.A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra, 38 (1976)29-44.