Linearly ordered group
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In abstract algebra a linearly ordered group is an ordered group <math>G</math> such that the order relation "<math>\leq</math>" is total. This means that the following statements hold for all <math>a,b,c\in G</math>:
- if <math>a\leq b</math> <math>b\leq a</math> then <math>a=b</math> (antisymmetry)
- if <math>a\leq b</math> and <math>b\leq c</math> then <math>a\leq c</math> (transitivity)
- <math>a\leq b</math> or <math>b\leq a</math>(totality)
- the order relation is translation invariant: if <math>a\leq b</math> then <math>a+c\leq b+c</math> and <math>c+a\leq c+b</math>.
In analogy with ordinary numbers, we call an element c of an ordered group positive if <math>0\leq c, c\neq 0</math>. The set of positive elements in a group is often denoted with <math>G_+</math>. For every element <math>a</math> of a linearly ordered group <math>G</math> either <math>a\in G_+</math>, or <math>-a\in G_+</math>, or <math>a=0</math> Template:Ref. If a linearly ordered group <math>G</math> is not trivial (i.e. <math>0</math> is not its only element), then <math>G_+</math> is infinite. Therefore, every nontrivial linearly ordered group is infinite Template:Ref.
If <math>a</math> is an element of a linearly ordered group <math>G</math>, then the absolute value of <math>a</math>, denoted by <math>|a| </math>, is defined to be:
- <math>|a| := \begin{cases} a, & \mbox{if } a \ge 0, \\ -a, & \mbox{otherwise}. \end{cases} </math>
If in addition the group <math>G</math> is abelian, then for any <math>a,b \in G</math> the triangle inequality is satisfied: <math>|a+b| \leq |a|+|b|</math> Template:Ref.
Otto Hölder showed that every linearly ordered group satisfying an Archimedean property is isomorphic to a subgroup of the additive group of real numbers.
The names below refer to the theorems formally verified by the IsarMathLib project.
- Template:Note OrderedGroup_ZF_1_L17
- Template:Note Linord_group_infinite
- Template:Note OrderedGroup_ZF_3_T1