linearly ordered set

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+.

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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