Linearly ordered group

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In abstract algebra a linearly ordered or totally ordered group is an ordered group $G$ such that the order relation " $\leq$ " is total. This means that the following statements hold for all $a,b,c\in G$ :

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

In analogy with ordinary numbers, we call an element c of an ordered group positive if $0\leq c, c\neq 0$ . The set of positive elements in a group is often denoted with $G +$ . For every element $a$ of a linearly ordered group $G$ either $a\in G_+$ , or $-a\in G_+$ , or $a = 0$ ^[1]. 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 ^[2].

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

$|a| := \begin{cases} a, & \mbox{if } a \ge 0, \\ -a, & \mbox{otherwise}. \end{cases}$

If in addition the group $G$ is abelian, then for any $a,b \in G$ the triangle inequality is satisfied: $|a+b| \leq |a|+|b|$ ^[3].

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.

^ OrdGroup_decomp
^ Linord_group_infinite
^ OrdGroup_triangle_ineq

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Category: Ordered groups

Linearly ordered group

From Wikipedia, the free encyclopedia

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