Linearly ordered group

Linearly ordered group

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":

* 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"+. [Note that the + is written as a subscript, to distinguish from "G"+ which includes the identity element. See e.g. [http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf IsarMathLib] , p. 285.]

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|:=egin{cases}a,& ext{if }ageqslant0,\-a,& ext{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.

References


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