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:

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, of the closure of an 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 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.