# Linearly ordered group

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

**left-ordered group**if*a*≤*b*implies*c+a*≤*c+b*,**right-ordered group**if*a*≤*b*implies*a+c*≤*b+c*,**bi-ordered group**if it is both left-ordered and right-ordered.

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 (not necessarily the familiar zero of the real numbers). The set of positive elements in a group is often denoted with *G*_{+}.[lower-alpha 1]

Elements of a linearly ordered group satisfy trichotomy: every element *a* of a linearly ordered group *G* is either positive (*a* ∈ *G*_{+}), negative (*−a* ∈ *G*_{+}), or zero (*a* = 0). If a linearly ordered group *G* is not trivial (i.e. 0 is not its only element), then *G*_{+} is infinite, since all multiples of a non-zero element are distinct.[lower-alpha 2] 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, 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 (Ghys 2001).

## Notes

- Note that the + is written as a subscript, to distinguish from
*G*^{+}which includes the identity element. See e.g. IsarMathLib, p. 344. - Formally, given any non-zero element
*c*(which we can assume to be positive, otherwise take*−c*) and natural number*k*we have , so by induction, given two natural numbers*k*<*l*, we have , so there is an injection from the natural numbers into*G*.

## References

- Levi, F.W. (1942), "Ordered groups.",
*Proc. Indian Acad. Sci.*,**A16**: 256–263 - Fuchs, László; Salce, Luigi (2001),
*Modules over non-Noetherian domains*, Mathematical Surveys and Monographs,**84**, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715 - Ghys, É. (2001), "Groups acting on the circle.",
*L´Eins. Math.*,**47**: 329–407