Editing Linearly ordered group (section)

===Archimedean ordered groups===

[[Otto Hölder]] showed that every [[Archimedean group]] (a bi-ordered group satisfying an [[Archimedean property]]) is [[isomorphism|isomorphic]] to a [[subgroup]] of the additive group of [[real number]]s, {{harv|Fuchs|Salce|2001|p=61}}.
If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the [[Dedekind–MacNeille completion|Dedekind completion]], <math>\widehat{G}</math> of the closure of a l.o. group under <math>n</math>th roots. We endow this space with the usual [[Topological_space#Definitions|topology]] of a linear order, and then it can be shown that for each <math>g\in\widehat{G}</math> the exponential maps <math>g^{\cdot}:(\mathbb{R},+)\to(\widehat{G},\cdot) :\lim_{i}q_{i}\in\mathbb{Q}\mapsto \lim_{i}g^{q_{i}}</math> 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 of a group|rank]]: which is related to the order type of the largest sequence of convex subgroups.