PlanetMath: ordered group

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ordered group

(Definition)

Definition 1. We say that the subsemigroup of the group (with the operation denoted multiplicatively) defines an order for the group , 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 is explicitly given by setting in :

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

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

Theorem 1 The order “

” defined by the subsemigroup

of the group

has the following properties.

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

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

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

Bibliography

1: EMIL ARTIN: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).
2: PAUL JAFFARD: Les systèmes d'idéaux. Dunod, Paris (1960).

"ordered group" is owned by pahio.

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Also defines:

ordered group equipped with zero

This object's parent.

Attachments:: proof of basic theorem about ordered groups (Proof) by rspuzio
isolated subgroup (Definition) by pahio
corollaries of basic theorem on ordered groups (Corollary) by rspuzio

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Cross-references: properties, order, pairwise disjoint, union, operation, group, subsemigroup
There are 13 references to this entry.

This is version 10 of ordered group, born on 2004-12-27, modified 2007-02-18.
Object id is 6595, canonical name is OrderedGroup.
Accessed 3003 times total.

Classification:

AMS MSC:	06A05 (Order, lattices, ordered algebraic structures :: Ordered sets :: Total order)
	20F60 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Ordered groups)

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