nLab ordered group

Idea

An ordered group is both a poset and a group in a compatible way. The concept applies directly to other constructs with group structure, such as ordered abelian groups, ordered vector spaces, etc. However, for ordered rings, ordered fields, and so on, additional compatibility conditions are required.

Definition

Let $G$ be a group (not necessarily commutative and yet written additively here), and let $\leq$ be a partial order on the underlying set of $G$. Then $(G,\leq)$ is an ordered group if this compatibility condition (translation invariance) holds:

• If $a \leq b$, then $a + c \leq b + c$ and $c + a \leq c + b$.

Remark

Ordered group are not equivalently group objects in Pos. The trouble is that requiring the inversion map $x \mapsto x^{-1}$ to preserve order (i.e., to be monotone, not antitone) is much too restrictive. Rather, an ordered group is a monoid object in the cartesian monoidal category $Pos$ which has the property that its underlying monoid in Set is a group.

If $G$ is an abelian group, then we have an ordered abelian group; in this case, only one direction of translation invariance needs to be checked.

It works just as well to talk of partially ordered monoids, using the same condition of translation invariance. Equivalently, an ordered monoid is a thin monoidal category, or a monoidal $(0,1)$-category.

Properties

The order $\leq$ is determined entirely by the group $G$ and the positive cone $G^+$:

$$G^+ \coloneqq \{x\colon G \;|\; 0 \leq x\} .$$

It's possible to define an ordered group in terms of the positive cone (by specifying precisely the conditions that the positive cone must satisfy); see positive cone for this.

However, this characterisation probably can't be made to work for ordered monoids (although I haven't checked for certain).

Examples

The underlying additive group of any ordered field is an ordered group.

In particular, the underlying additive group of the field $\mathbb{R}$ of real numbers is an ordered group.

Although the field $\mathbb{C}$ of complex numbers is not an ordered field (since it is not strictly totally ordered), its underlying additive group is still an ordered group (where $a \leq b$ means that $b - a$ is a nonnegative real number).

Given a topological vector space $V$, we often consider its dual vector space $V^*$, consisting of the continuous linear maps from $V$ to its base field, which is usually either $\mathbb{R}$ or $\mathbb{C}$. This inherits a partial order from the target field, and then the underlying additive group is an ordered group; in fact, we have an ordered algebra?. (This is the main sort of example that I know of, but that probably just reflects my own limited knowledge.)

More generally, if $V$ is any set, $G$ is any ordered group, and $F$ is any collection of functions from $V$ to $G$, as long as $F$ is a subgroup of the group of all functions from $V$ to $G$, then $F$ is an ordered group.

Non-abelian examples include free groups and torsion-free nilpotent groups. The basics of the theory for both abelian and nonabelian ordered groups can be found in Birkhoff’s Lattice Theory.

Related concepts

• lattice-ordered group

• Tarski group

• protoring

References

• English Wikipedia: Partially ordered group.

The stable category of preordered groups is studied in

• Aline Michel?, The stable category of preordered groups, 2023. [arXiv:2309.11953]

This categorical properties of preordered groups was also studied in

• Maria Manuel Clementino, Nelson Martins-Ferreira?, and Andrea Montoli?, On the categorical behaviour of preordered groups, Journal of Pure and Applied Algebra, Vol. 223, No. 10, 2019. [doi:doi.org/10.1016/j.jpaa.2019.01.006]