In abstract algebra a linearly ordered or totally ordered group is a group G equipped with a total order "≤", that is translation-invariant. This may have different meanings. Let a, b, c ∈ G, we say that (G, ≤) is a
In analogy with ordinary numbers, we call an element c of an ordered group positive if 0 ≤ c and c ≠ 0, where "0" here denotes the identity element of the group (not necessarily the familiar zero of the real numbers). 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|.
Examples
Any totally ordered group is torsion-free. Conversely, F. W. Levi showed that an abelian group admits a linear order if and only if it is torsion free (Levi 1942).
Otto Hölder showed that every Archimedean group (a bi-ordered group satisfying an Archimedean property) is isomorphic to a subgroup of the additive group of real numbers, (Fuchs & Salce 2001, p. 61). If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion,
A large source of examples of left-orderable groups comes from groups acting on the real line by order preserving homeomorphisms. Actually, for countable groups, this is known to be a characterization of left-orderability, see for instance (Ghys 2001).