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In [[abstract algebra]] a '''linearly ordered''' or '''totally ordered group''' is a [[group (mathematics)|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''
* '''bi-ordered group''' if ''a'' ≤ ''b'' implies ''c a'' ≤ ''c b'' and ''a c'' ≤ ''b c''


In analogy with ordinary numbers, we call an element ''c'' of an ordered group '''positive''' if 0&nbsp;≤&nbsp;''c'' and ''c''&nbsp;≠&nbsp;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''<sub>+</sub>.<ref>Note that the + is written as a subscript, to distinguish from ''G''<sup>+</sup> which includes the identity element. See e.g. [http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf IsarMathLib], p. 344.</ref>
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For every element ''a'' of a linearly ordered group ''G'' either ''a''&nbsp;∈&nbsp;''G''<sub>+</sub>, or '' -a'' &nbsp;∈&nbsp;''G''<sub>+</sub>, or ''a''&nbsp;=&nbsp;0. If a linearly ordered group ''G'' is not trivial (i.e. 0 is not its only element), then ''G''<sub>+</sub> 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:
 
:<math>|a|:=\begin{cases}a, & \text{if }a\geqslant0,\\ -a, & \text{otherwise}.\end{cases}</math>
 
If in addition the group ''G'' is [[abelian group|abelian]], then for any ''a'',&nbsp;''b''&nbsp;∈&nbsp;''G'' the [[triangle inequality]] is satisfied: |''a''&nbsp;+&nbsp;''b''| ≤&nbsp;|''a''|&nbsp;+&nbsp;|''b''|.
 
==Examples==
 
 
Any totally ordered group is [[Torsion (algebra)|torsion-free]]. Conversely, [[F. W. Levi]] showed that an [[abelian group]] admits a linear order if and only if it is torsion free {{harv|Levi|1942}}.
 
[[Otto Hölder]] showed that every 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 completion, <math>\widehat{G}</math> of the closure of an  l.o. group under <math>n</math>th roots. We endow this space with the usual 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 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 {{harv|Ghys|2001}}.
 
==See also==
*[[Cyclically ordered group]]
 
==Notes==
<references />
 
==References==
 
*{{Citation | last1=Levi | first1=F.W. | title=Ordered groups. | journal=Proc. Indian Acad. Sci. | year=1942 | volume=A16 | pages=256–263}}
 
*{{Citation | last1=Fuchs | first1=László | last2=Salce | first2=Luigi | title=Modules over non-Noetherian domains | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1963-0 | id={{MathSciNet | id = 1794715}} | year=2001 | volume=84}}
 
*{{Citation | last1=Ghys | first1=É. | title=Groups acting on the circle. | journal=L´Eins. Math. | year=2001 | volume=47 | pages=329–407}}
 
[[Category:Ordered groups]]

Revision as of 10:32, 4 March 2014


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