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<b\\,\\,\\Leftrightarrow\\,\\,ab^{-1}\\in S

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.
For all $a,\\,b\\in G$ , exactly one of the conditions $a<b,\\,\\,a=b,\\,\\,b<a$ holds.
2.
$a<b\\,\\land\\,b<c\\,\\,\\Rightarrow\\,\\,a<c$
3.
$a<b\\,\\,\\Rightarrow\\,\\,ac<bc\\,\\land\\,ca<cb$
4.
$a<b\\,\\land\\,c<d\\,\\,\\Rightarrow\\,\\,ac<bd$
5.
$a<b\\,\\,\\Leftrightarrow\\,\\,b^{-1}<a^{-1}$
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

•
$0a=a0=0\\quad\\forall a\\in G,$
•
$0<a\\quad\\forall a\\in G^{*}.$

Cf. 7 in examples of semigroups.

References

1 Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).
2 Paul Jaffard: Les systèmes d’idéaux. Dunod, Paris (1960).