nLab ordered group

Ordered groups



(0,1)-Category theory

Ordered groups


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.


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

  • If aba \leq b, then a+cb+ca + c \leq b + c and c+ac+bc + a \leq c + b.

An ordered group is not the same thing as a group object in PosPos. The trouble is that requiring the inversion map xx 1x \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 PosPos which has the property that its underlying monoid in SetSet is a group.

If GG 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)(0,1)-category.


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

G +{x:G|0x}. 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).


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 aba \leq b means that bab - a is a nonnegative real number).

Given a topological vector space VV, we often consider its dual vector space V *V^*, consisting of the continuous linear maps from VV 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 VV is any set, GG is any ordered group, and FF is any collection of functions from VV to GG, as long as FF is a subgroup of the group of all functions from VV to GG, then FF 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.


The stable category of preordered groups is studied in

This categorical properties of preordered groups was also studied in

Last revised on February 23, 2024 at 20:33:12. See the history of this page for a list of all contributions to it.