Editing Linearly ordered group

{{Short description|Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb}}
In [[mathematics]], specifically [[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. We say that (''G'', ≤) is a:

* '''left-ordered group''' if ≤ is left-invariant, that is ''a''&nbsp;≤&nbsp;''b'' implies ''ca''&nbsp;≤&nbsp;''cb'' for all ''a'',&nbsp;''b'',&nbsp;''c'' in ''G'',
* '''right-ordered group''' if ≤ is right-invariant, that is  ''a''&nbsp;≤&nbsp;''b'' implies ''ac''&nbsp;≤&nbsp;''bc'' for all ''a'',&nbsp;''b'',&nbsp;''c'' in ''G'',
* '''bi-ordered group''' if ≤ is bi-invariant, that is it is both left- and right-invariant. 

A group ''G'' is said to be '''left-orderable''' (or '''right-orderable''', or '''bi-orderable''') if there exists a left- (or right-, or bi-) invariant order on ''G''. A simple necessary condition for a group to be left-orderable is to have no elements of finite order; however this is not a sufficient condition. It is equivalent for a group to be left- or right-orderable; however there exist left-orderable groups which are not bi-orderable. 

== Further definitions ==

{{more footnotes needed|section|date=July 2024}}

In this section <math>\le</math> is a left-invariant order on a group <math>G</math> with [[identity element]] <math>e</math>. All that is said applies to right-invariant orders with the obvious modifications. Note that <math>\le</math> being left-invariant is equivalent to the order <math>\le'</math> defined by <math>g \le' h</matH> if and only if <math>h^{-1} \le g^{-1}</math> being right-invariant. In particular a group being left-orderable is the same as it being right-orderable. 

In analogy with ordinary numbers we call an element <math>g \not= e</math> of an ordered group '''positive''' if <math>e \le g</math>. The set of positive elements in an ordered group is called the '''positive cone''', it is often denoted with <math>G_+</math>; the slightly different notation <math>G^+</math> is used for the positive cone together with the identity element.{{sfn|Deroin|Navas|Rivas|2014|loc=1.1.1}} 

The positive cone <math>G_+</math> characterises the order <math>\le</math>; indeed, by left-invariance we see that <math>g \le h</math> if and only if <math>g^{-1} h \in G_+</math>. In fact a left-ordered group can be defined as a group <math>G</math> together with a subset <math>P</math> satisfying the two conditions that:
#for <math>g, h \in P</math> we have also <math>gh \in P</math>;
#let <math>P^{-1} = \{g^{-1}, g \in P\}</math>, then <math>G</math> is the [[disjoint union]] of <math>P, P^{-1}</math> and <math>\{e\}</math>. 
The order <math>\le_P</math> associated with <math>P</math> is defined by <math>g \le_P h \Leftrightarrow g^{-1} h \in P</math>; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of <math>\le_P</math> is <math>P</math>.

The left-invariant order <math>\le</math> is bi-invariant if and only if it is conjugacy invariant, that is if <math>g \le h</math> then for any <math>x \in G</math> we have <math>xgx^{-1} \le xhx^{-1}</math> as well. This is equivalent to the positive cone being stable under [[inner automorphism]]s. 


If <math>a \in G</math>{{cn|date=July 2024}}, then the '''absolute value''' of <math>a</math>, denoted by <math>|a|</math>, is defined to be: <math display=block>|a|:=\begin{cases}a, & \text{if }a \ge 0,\\ -a, & \text{otherwise}.\end{cases}</math>
If in addition the group <math>G</math> is [[abelian group|abelian]], then for any <math>a, b \in G</math> a [[triangle inequality]] is satisfied: <math>|a+b| \le |a|+|b|</math>.

==Examples==

Any left- or right-orderable group is [[Torsion (algebra)|torsion]]-free, that is it contains no elements of finite order besides the identity. Conversely, [[F. W. Levi]] showed that a torsion-free [[abelian group]] is bi-orderable;{{sfn|Levi|1942}} this is still true for [[nilpotent group]]s{{sfn|Deroin|Navas|Rivas|2014|loc=1.2.1}} but there exist torsion-free, [[finitely presented group]]s which are not left-orderable. 

===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.

===Other examples===

[[Free group]]s are left-orderable. More generally this is also the case for [[right-angled Artin group]]s.<ref>{{cite journal |last1=Duchamp |first1=Gérard |last2=Thibon |first2=Jean-Yves |date= 1992|title=Simple orderings for free partially commutative groups |url= |journal=International Journal of Algebra and Computation |volume=2 |issue=3 |pages=351–355 |doi=10.1142/S0218196792000219 |zbl=0772.20017 |access-date=}}</ref> [[Braid group]]s are also left-orderable.<ref>{{cite book |last1=Dehornoy |first1=Patrick |last2=Dynnikov |first2=Ivan |last3=Rolfsen |first3=Dale |last4=Wiest |first4=Bert | author-link= |date= 2002|title=Why are braids orderable? |url= |location=Paris |publisher=Société Mathématique de France |page=xiii + 190 |isbn=2-85629-135-X}}</ref>

The group given by the presentation <math>\langle a, b | a^2ba^2b^{-1}, b^2ab^2a^{-1}\rangle</math> is torsion-free but not left-orderable;{{sfn|Deroin|Navas|Rivas|2014|loc=1.4.1}} note that it is a 3-dimensional [[crystallographic group]] (it can be realised as the group generated by two glided half-turns with orthogonal axes and the same translation length), and it is the same group that was proven to be a counterexample to the [[Kaplansky conjectures|unit conjecture]]. More generally the topic of orderability of 3--manifold groups is interesting for its relation with various topological invariants.<ref>{{cite journal |last1=Boyer |first1=Steven |last2=Rolfsen |first2=Dale| last3=Wiest| first3=Bert |date=2005 |title=Orderable 3-manifold groups |doi=10.5802/aif.2098 |journal=Annales de l'Institut Fourier |volume=55 |issue=1 |pages=243–288 | zbl=1068.57001|doi-access=free |arxiv=math/0211110 }}</ref> There exists a 3-manifold group which is left-orderable but not bi-orderable<ref>
{{cite journal |last=Bergman |first=George |date=1991 |title=Right orderable groups that are not locally indicable |url= |journal=Pacific Journal of Mathematics |volume=147 |issue=2 |pages=243–248 |doi=10.2140/pjm.1991.147.243 |zbl=0677.06007|doi-access=free }}</ref> (in fact it does not satisfy the weaker property of being locally indicable). 

Left-orderable groups have also attracted interest from the perspective of [[dynamical system]]s as it is known that a countable group is left-orderable if and only if it acts on the real line by homeomorphisms.{{sfn|Deroin|Navas|Rivas|2014|loc=Proposition 1.1.8}} Non-examples related to this paradigm are [[Lattice (discrete subgroup)|lattices]] in higher rank [[Lie group]]s; it is known that (for example) finite-index subgroups in <math>\mathrm{SL}_n(\mathbb Z)</math> are not left-orderable;<ref>{{cite journal |last=Witte |first=Dave |date=1994 |title=Arithmetic groups of higher \(\mathbb{Q}\)-rank cannot act on \(1\)-manifolds |journal=Proceedings of the American Mathematical Society |volume=122 |issue=2 |pages=333–340 |doi=10.2307/2161021 |jstor=2161021 | zbl=0818.22006}}</ref> a wide generalisation of this has been recently announced.<ref>{{cite arXiv |last1=Deroin |first1=Bertrand |last2=Hurtado |first2=Sebastian |eprint=2008.10687 |title=Non left-orderability of lattices in higher rank semi-simple Lie groups |class=math.GT |date=2020 }}</ref>

==See also==
*[[Cyclically ordered group]]
*[[Hahn embedding theorem]]
*[[Partially ordered group]]

==Notes==
{{reflist|group=lower-alpha}}
{{reflist}}

==References==

*{{cite arXiv |last1=Deroin |first1=Bertrand |last2=Navas |first2=Andrés |last3=Rivas |first3=Cristóbal |author-link= |eprint=1408.5805 |title=Groups, orders and dynamics |class=math.GT |date=2014 }}
*{{Citation | last1=Levi | first1=F.W. | title=Ordered groups. | journal=Proc. Indian Acad. Sci. | year=1942 | volume=A16 | issue=4 | pages=256–263| doi=10.1007/BF03174799 | s2cid=198139979 }}
*{{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 |mr=1794715 | year=2001 | volume=84}}
*{{Citation | last1=Ghys | first1=É. | title=Groups acting on the circle. | journal=L'Enseignement Mathématique | year=2001 | volume=47 | pages=329–407}}

[[Category:Ordered groups]]