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 (G, ≤) is a:

• left-ordered group if a ≤ b implies c+a ≤ c+b for all a, b, c in G,
• right-ordered group if a ≤ b implies a+c ≤ b+c for all a, b, c in G,
• 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:

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

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

See also

• Cyclically ordered group
• Hahn embedding theorem
• Partially ordered group

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.
2. Formally, given any non-zero element c (which we can assume to be positive, otherwise take −c) and natural number k we have ${\displaystyle kc<(k+1)c}$, so by induction, given two natural numbers k < l, we have ${\displaystyle kc<lc}$, so there is an injection from the natural numbers into G.

References

• Levi, F.W. (1942), "Ordered groups.", Proc. Indian Acad. Sci., A16 (4): 256–263, doi:10.1007/BF03174799
• 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'Enseignement Mathématique, 47: 329–407