ordered group

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 ring?s, ordered fields, and so on, additional compatibility conditions are required.

Let $G$ be a group (written additively but not necessarily commutative), 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 $c + a \leq c + b$.

More slickly, an ordered group is (up to equivalence) a thin groupal category? (a groupal $(0,1)$-category).

An ordered group is *not* the same thing as a group object 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.

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).

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 *linearly* 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.

All of these examples are commutative, but Wikipedia defines the concept for noncommutative groups as well, so presumably somebody has done something with those too.

- English Wikipedia: Partially ordered group.

Revised on May 21, 2017 22:13:21
by Todd Trimble
(67.81.95.215)