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
- 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:
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.