Totally 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

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|.

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 an 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 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.

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