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

•
${a}^{1}Sa\subseteq S\mathit{\hspace{1em}}\forall a\in G,$

•
$G=S\cup \{1\}\cup {S}^{1}$ where ${S}^{1}=\{{s}^{1}:s\in S\}$ and the members of the union are pairwise disjoint.
The order “$$” of the group $G$ is explicitly given by setting in $G$:
$$ 
Then we speak of the ordered group $$, or simply $G$.
Theorem 1.
The order “$$” 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 $$ holds.

2.
$$

3.
$$

4.
$$

5.
$$

6.
$$
Definition 2. The set $G$ is an ordered group equipped with zero 0, if the set ${G}^{*}$ of its elements distinct from its element 0 forms an ordered group $$ and if

•
$0a=a0=0\mathit{\hspace{1em}}\forall a\in G,$

•
$$
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).
Title  ordered group 

Canonical name  OrderedGroup 
Date of creation  20130322 14:54:36 
Last modified on  20130322 14:54:36 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  16 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 06A05 
Classification  msc 20F60 
Related topic  KrullValuation 
Related topic  PartiallyOrderedGroup 
Related topic  PraeclarumTheorema 
Defines  ordered group equipped with zero 