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[parent] ordered group (Definition)

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

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

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

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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See Also: Krull valuation, partially ordered group

Also defines:  ordered group equipped with zero

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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 MSC06A05 (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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