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).

Title	ordered group
Canonical name	OrderedGroup
Date of creation	2013-03-22 14:54:36
Last modified on	2013-03-22 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