# 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:

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.

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

A *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]}

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