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

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₊.^[1]

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:

$|a|:={\begin{cases}a,&{\text{if }}a\geqslant 0,\\-a,&{\text{otherwise}}.\end{cases}}$

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

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, $\widehat{G}$ 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.

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).

See also

• Cyclically ordered group
• Hahn embedding theorem

Notes

1. Note that the + is written as a subscript, to distinguish from G⁺ which includes the identity element. See e.g. IsarMathLib, p. 344.

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