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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 $a\leq b$, then
1. $ac\leq bc$, and
2. $ca\leq cb$,
for any $c\in G$. The two conditions are equivalent to the one condition $cad\leq cbd$ 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 $a\leq b$, then $b^{{1}}\leq a^{{1}}$. To see this, left multiply by the first inequality by $a^{{1}}$ on both sides to obtain $e\leq a^{{1}}b$. Then right multiply the resulting inequality on both sides by $b^{{1}}$ to obtain the desired inequality: $b^{{1}}\leq a^{{1}}$.

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 $b\leq c$, then $L_{a}(b)=ab\leq ac=L_{a}(c)$.

A element $a$ in a pogroup $G$ is said to be positive if $e\leq a$, 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 $a\leq c$ and $b\leq c$, so that $ac^{{1}}b\leq a$ and $ac^{{1}}b\leq b$ as well.

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

(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 $a^{n}\leq b$ for all $n\in\mathbb{Z}$, then $a=e$. Equivalently, if $a\neq e$, then for any $b\in G$, there is some $n\in\mathbb{Z}$ such that $b<a^{n}$. 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 $r<n$. To see this, pick $b=r$, and $a=1$.
(e) A pogroup is said to be integrally closed if $a^{n}\leq b$ for all $n\geq 1$, then $a\leq e$. An integrally closed group is Archimedean: if $a^{n}\leq b$ for all $n\in\mathbb{Z}$, then $a\leq e$ and $e\leq b$. Since we also have $(a^{{1}})^{{n}}\leq b$ for all $n<0$, this implies $a^{{1}}\leq e$, or $e\leq a$. Hence $a=e$. In fact, an directed integrally closed group is an Abelian pogroup.

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.
Mathematics Subject Classification
06F05 no label found06F20 no label found06F15 no label found20F60 no label found Forums
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