partially ordered group
A partially ordered group is a group $G$ that is a poset at the same time, such that if $a,b\in G$ and $a\le b$, then

1.
$ac\le bc$, and

2.
$ca\le cb$,
for any $c\in G$. The two conditions are equivalent^{} to the one condition $cad\le cbd$ for all $c,d\in G$. A partially ordered group is also called a pogroup for short.
Remarks.

•
One of the immediate properties of a pogroup is this: if $a\le b$, then ${b}^{1}\le {a}^{1}$. To see this, left multiply by the first inequality by ${a}^{1}$ on both sides to obtain $e\le {a}^{1}b$. Then right multiply the resulting inequality on both sides by ${b}^{1}$ to obtain the desired inequality: ${b}^{1}\le {a}^{1}$.

•
If can be seen that for every $a\in G$, the automorphisms^{} ${L}_{a},{R}_{a}:G\to G$ also preserve order, and hence are order automorphisms as well. For instance, if $b\le c$, then ${L}_{a}(b)=ab\le ac={L}_{a}(c)$.

•
A element $a$ in a pogroup $G$ is said to be positive if $e\le a$, where $e$ is the identity element^{} of $G$. The set of positive elements in $G$ is called the positive cone^{} of $G$.

•
(special pogroups)

(a)
A pogroup whose underlying poset is a directed set^{} is called a directed group.

*
If $G$ is a directed group, then $G$ is also a filtered set: if $a,b\in G$, then there is a $c\in G$ such that $a\le c$ and $b\le c$, so that $a{c}^{1}b\le a$ and $a{c}^{1}b\le b$ as well.

*
Also, if $G$ is directed, then $G=\u27e8{G}^{+}\u27e9$: for any $x\in G$, let $a$ be the upper bound of $\{x,e\}$ and let $b=a{x}^{1}$. Then $e\le b$ and $x={a}^{1}b\in \u27e8{G}^{+}\u27e9$.

*

(b)
A pogroup whose underlying poset is a lattice^{} is called a lattice ordered group, or an lgroup.

(c)
If the partial order^{} on a pogroup $G$ is a linear order, then $G$ is called a totally ordered group, or simply an ordered group.

(d)
A pogroup is said to be Archimedean^{} if ${a}^{n}\le b$ for all $n\in \mathbb{Z}$, then $a=e$. Equivalently, if $a\ne e$, then for any $b\in G$, there is some $n\in \mathbb{Z}$ such that $$. This is a generalization^{} of the Archimedean property on the reals: if $r\in \mathbb{R}$, then there is some $n\in \mathbb{N}$ such that $$. To see this, pick $b=r$, and $a=1$.

(e)
A pogroup is said to be integrally closed^{} if ${a}^{n}\le b$ for all $n\ge 1$, then $a\le e$. An integrally closed group is Archimedean: if ${a}^{n}\le b$ for all $n\in \mathbb{Z}$, then $a\le e$ and $e\le b$. Since we also have ${({a}^{1})}^{n}\le b$ for all $$, this implies ${a}^{1}\le e$, or $e\le a$. Hence $a=e$. In fact, an directed integrally closed group is an Abelian^{} pogroup.

(a)

•
Since the definition above does not involve any specific group axioms, one can more generally introduce partial ordering on a semigroup^{} in the same fashion. The result is called a partially ordered semigroup, or a posemigroup for short. A lattice ordered semigroup is defined similarly.
Title  partially ordered group 
Canonical name  PartiallyOrderedGroup 
Date of creation  20130322 16:42:25 
Last modified on  20130322 16:42:25 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  14 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06F05 
Classification  msc 06F20 
Classification  msc 06F15 
Classification  msc 20F60 
Synonym  pogroup 
Synonym  lgroup 
Synonym  Archimedean pogroup 
Synonym  integrally closed pogroup 
Synonym  posemigroup 
Synonym  latticeordered group 
Synonym  lsemigroup 
Related topic  OrderedGroup 
Defines  directed group 
Defines  positive element 
Defines  positive cone 
Defines  lattice ordered group 
Defines  Archimedean partially ordered group 
Defines  integrally closed group 
Defines  integrally closed partially ordered group 
Defines  partially ordered semigroup 
Defines  lattice ordered semigroup 
Defines  Archimedean 