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 po-group for short.

Remarks.

• One of the immediate properties of a po-group 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 po-group $G$ 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 po-groups)
  1. A po-group 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 $ac^{-1}b\le a$ and $ac^{-1}b\le b$ as well.
     • Also, if $G$ is directed, then $G=\langle G^+\rangle$ : for any $x\in G$ , let $a$ be the upper bound of $\lbrace x,e\rbrace$ and let $b=ax^{-1}$ . Then $e\le b$ and $x=a^{-1}b\in \langle G^+\rangle$ .
  2. A po-group whose underlying poset is a lattice is called a lattice ordered group, or an l-group.
  3. If the partial order on a po-group $G$ is a linear order, then $G$ is called a totally ordered group, or simply an ordered group.
  4. A po-group 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 $b<a^n$ . 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 $a=1$ .
  5. A po-group 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 $n<0$ , this implies $a^{-1}\le e$ , or $e\le a$ . Hence $a=e$ . In fact, an directed integrally closed group is an Abelian po-group.
• 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 po-semigroup for short. A lattice ordered semigroup is defined similarly.