# 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$_j$, 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 K$_0$(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 $G$ is called Archimedean when for any $a,b\in G$, if $e\leq a\leq b$ and $a^{n}\leq b$ for all $n\geq 1$ then $a=e$. Equivalently, when $a\neq e$, then for any $b\in G$, there is some $n\in \mathbb {Z}$ 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.