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In abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the order relation "≤" is total. This means that the following statements hold for all a,b,c ∈ G:## References

- if a ≤ b and b ≤ a then a = b (antisymmetry)
- if a ≤ b and b ≤ c then a ≤ c (transitivity)
- a ≤ b or b ≤ a (totality)
- the order relation is translation invariant: if a ≤ b then a + c ≤ b + c and c + a ≤ c + b.

In analogy with ordinary numbers, we call an element c of an ordered group positive if 0 ≤ c and c ≠ 0. The set of positive elements in a group is often denoted with G_{+}.

For every element a of a linearly ordered group G either a ∈ G_{+}, or −a ∈ G_{+}, or a = 0. If a linearly ordered group G is not trivial (i.e. 0 is not its only element), then G_{+} is infinite. Therefore, every nontrivial linearly ordered group is infinite.

If a is an element of a linearly ordered group G, then the absolute value of a, denoted by |a|, is defined to be:

- $|a|:=begin\{cases\}a,\&text\{if\; \}ageqslant0,-a,\&text\{otherwise\}.end\{cases\}$

If in addition the group G is abelian, then for any a,b ∈ G the triangle inequality is satisfied: |a + b| ≤ |a| + |b|.

Otto Hölder showed that every linearly ordered group satisfying an Archimedean property is isomorphic to a subgroup of the additive group of real numbers.

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Last updated on Monday September 29, 2008 at 05:36:08 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday September 29, 2008 at 05:36:08 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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