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, 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:

$|a|:=\begin{cases}a, & \text{if }a\geqslant0,\\ -a, & \text{otherwise}.\end{cases}$

If in addition the group G is abelian, then for any a, b ∈ G the triangle inequality is satisfied: |a + b| ≤ |a| + |b|.

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 linearly 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, $\widehat{G}$ of the closure of an l.o. group under $n$th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each $g\in\widehat{G}$ the exponential maps $g^{\cdot}:(\mathbb{R},+)\to(\widehat{G},\cdot) :\lim_{i}q_{i}\in\mathbb{Q}\mapsto \lim_{i}g^{q_{i}}$ are well defined order preserving/reversing, topological group isomorphisms.

Completing an 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.

See also

• Cyclically ordered group

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, MR1794715