Linearly ordered group


In mathematics, specifically 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. We say that is a:
Note that G need not be abelian, even though we use additive notation for the group operation.

Definitions

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. The set of positive elements in a group is often denoted with G+.
Elements of a linearly ordered group satisfy trichotomy: every element a of a linearly ordered group G is either positive, negative, or zero. If a linearly ordered group G is not trivial, then G+ is infinite, since all multiples of a non-zero element are distinct. 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.
Otto Hölder showed that every Archimedean group is isomorphic to a subgroup of the additive group of real numbers,.
If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion, of the closure of a l.o. group under th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each the exponential maps are well defined order preserving/reversing, topological group isomorphisms. Completing a l.o. group can be difficult in the non-Archimedean case. In these cases, one may classify a group by its rank: which is related to the order type of the largest sequence of convex subgroups.
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.