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\begin{document}

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\section*{ordered group}

\hypertarget{ordered_groups}{}\section*{{Ordered groups}}\label{ordered_groups}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 ring]]s, [[ordered fields]], and so on, additional compatibility conditions are required.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $G$ be a [[group]] (written additively but not necessarily [[abelian group|commutative]]), and let $\leq$ be a [[partial order]] on the [[underlying set]] of $G$. Then $(G,\leq)$ is an \textbf{ordered group} if this compatibility condition (\emph{translation invariance}) holds:

\begin{itemize}%
\item If $a \leq b$, then $a + c \leq b + c$ and $c + a \leq c + b$.

\end{itemize}
More slickly, an ordered group is (up to [[equivalence of categories|equivalence]]) a [[thin category|thin]] [[groupal category]] (a groupal $(0,1)$-[[(0,1)-category|category]]).

\begin{remark}
\label{}\hypertarget{}{}
An ordered group is \emph{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 [[stuff, structure, property|property]] that its underlying monoid in $Set$ is a group.

\end{remark}
If $G$ is an [[abelian group]], then we have an \textbf{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 \textbf{ordered monoid} is a thin [[monoidal category]], or a monoidal $(0,1)$-category.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

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

\begin{displaymath}
G^+ \coloneqq \{x\colon G \;|\; 0 \leq x\} .
\end{displaymath}
It's possible to \emph{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).

\hypertarget{examples}{}\subsection*{{Examples}}\label{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 \emph{[[linear order|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 map|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.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item English Wikipedia: \href{https://en.wikipedia.org/wiki/Partially_ordered_group}{Partially ordered group}.

\end{itemize}
[[!redirects ordered group]] [[!redirects ordered groups]] [[!redirects partially ordered group]] [[!redirects partially ordered groups]] [[!redirects partially-ordered group]] [[!redirects partially-ordered groups]]

[[!redirects ordered abelian group]] [[!redirects ordered abelian groups]] [[!redirects partially ordered abelian group]] [[!redirects partially ordered abelian groups]] [[!redirects partially-ordered abelian group]] [[!redirects partially-ordered abelian groups]]

[[!redirects ordered monoid]] [[!redirects ordered monoids]] [[!redirects partially ordered monoid]] [[!redirects partially ordered monoids]] [[!redirects partially-ordered monoid]] [[!redirects partially-ordered monoids]]



\end{document}