# ordered group

Definition 1.  We say that the subsemigroup $S$ of the group $G$ (with the operation denoted multiplicatively) defines an $G$, if

• $a^{-1}Sa\subseteq S\quad\forall a\in G,$

• $G=S\cup\{1\}\cup S^{-1}$   where  $S^{-1}=\{s^{-1}:\,s\in S\}$  and the members of the union are pairwise disjoint.

The order “$<$” of the group $G$ is explicitly given by setting in $G$:

 $a

Then we speak of the ordered group$(G,\,<)$,  or simply $G$.

###### Theorem 1.

The order “$<$” defined by the subsemigroup $S$ of the group $G$ has the following properties.

1. 1.

For all  $a,\,b\in G$, exactly one of the conditions   $a   holds.

2. 2.

$a

3. 3.

$a

4. 4.

$a

5. 5.

$a

6. 6.

$a<1\,\,\Leftrightarrow\,\,a\in S$

Definition 2.  The set $G$ is an ordered group equipped with zero 0, if the set $G^{*}$ of its elements distinct from its element 0 forms an ordered group  $(G^{*},\,<)$  and if

• $0a=a0=0\quad\forall a\in G,$

• $0

## References

• 1 Emil Artin: Theory of Algebraic Numbers.  Lecture notes.  Mathematisches Institut, Göttingen (1959).
• 2 Paul Jaffard: Les systèmes d’idéaux.  Dunod, Paris (1960).
Title ordered group OrderedGroup 2013-03-22 14:54:36 2013-03-22 14:54:36 pahio (2872) pahio (2872) 16 pahio (2872) Definition msc 06A05 msc 20F60 KrullValuation PartiallyOrderedGroup PraeclarumTheorema ordered group equipped with zero