• Source: Linearly ordered group

In mathematics, specifically 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. We say that (G, ≤) is a:

left-ordered group if ≤ is left-invariant, that is a ≤ b implies ca ≤ cb for all a, b, c in G,
right-ordered group if ≤ is right-invariant, that is a ≤ b implies ac ≤ bc for all a, b, 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



In this section



≤


{\displaystyle \leq }

is a left-invariant order on a group



G


{\displaystyle G}

with identity element



e


{\displaystyle e}

. All that is said applies to right-invariant orders with the obvious modifications. Note that



≤


{\displaystyle \leq }

being left-invariant is equivalent to the order




≤
′



{\displaystyle \leq '}

defined by



g

≤
′

h


{\displaystyle g\leq 'h}

if and only if




h

−
1


≤

g

−
1




{\displaystyle h^{-1}\leq g^{-1}}

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



g
≠
e


{\displaystyle g\not =e}

of an ordered group positive if



e
≤
g


{\displaystyle e\leq g}

. The set of positive elements in an ordered group is called the positive cone, it is often denoted with




G

+




{\displaystyle G_{+}}

; the slightly different notation




G

+




{\displaystyle G^{+}}

is used for the positive cone together with the identity element.
The positive cone




G

+




{\displaystyle G_{+}}

characterises the order



≤


{\displaystyle \leq }

; indeed, by left-invariance we see that



g
≤
h


{\displaystyle g\leq h}

if and only if




g

−
1


h
∈

G

+




{\displaystyle g^{-1}h\in G_{+}}

. In fact a left-ordered group can be defined as a group



G


{\displaystyle G}

together with a subset



P


{\displaystyle P}

satisfying the two conditions that:

for



g
,
h
∈
P


{\displaystyle g,h\in P}

we have also



g
h
∈
P


{\displaystyle gh\in P}

;
let




P

−
1


=
{

g

−
1


,
g
∈
P
}


{\displaystyle P^{-1}=\{g^{-1},g\in P\}}

, then



G


{\displaystyle G}

is the disjoint union of



P
,

P

−
1




{\displaystyle P,P^{-1}}

and



{
e
}


{\displaystyle \{e\}}

.
The order




≤

P




{\displaystyle \leq _{P}}

associated with



P


{\displaystyle P}

is defined by



g

≤

P


h
⇔

g

−
1


h
∈
P


{\displaystyle g\leq _{P}h\Leftrightarrow g^{-1}h\in P}

; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of




≤

P




{\displaystyle \leq _{P}}

is



P


{\displaystyle P}

.
The left-invariant order



≤


{\displaystyle \leq }

is bi-invariant if and only if it is conjugacy invariant, that is if



g
≤
h


{\displaystyle g\leq h}

then for any



x
∈
G


{\displaystyle x\in G}

we have



x
g

x

−
1


≤
x
h

x

−
1




{\displaystyle xgx^{-1}\leq xhx^{-1}}

as well. This is equivalent to the positive cone being stable under inner automorphisms.

If



a
∈
G


{\displaystyle a\in G}

, then the absolute value of



a


{\displaystyle a}

, denoted by




|

a

|



{\displaystyle |a|}

, is defined to be:




|

a

|

:=


{



a
,



if

a
≥
0
,




−
a
,



otherwise

.








{\displaystyle |a|:={\begin{cases}a,&{\text{if }}a\geq 0,\\-a,&{\text{otherwise}}.\end{cases}}}


If in addition the group



G


{\displaystyle G}

is abelian, then for any



a
,
b
∈
G


{\displaystyle a,b\in G}

a triangle inequality is satisfied:




|

a
+
b

|

≤

|

a

|

+

|

b

|



{\displaystyle |a+b|\leq |a|+|b|}

.




Examples


Any left- or right-orderable group is 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; this is still true for nilpotent groups but there exist torsion-free, finitely presented groups which are not left-orderable.




= Archimedean ordered groups

=
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,






G
^





{\displaystyle {\widehat {G}}}

of the closure of a l.o. group under



n


{\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



g
∈



G
^





{\displaystyle g\in {\widehat {G}}}

the exponential maps




g

⋅


:
(

R

,
+
)
→
(



G
^



,
⋅
)
:

lim

i



q

i


∈

Q

↦

lim

i



g


q

i






{\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 a 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.




= Other examples

=
Free groups are left-orderable. More generally this is also the case for right-angled Artin groups. Braid groups are also left-orderable.
The group given by the presentation



⟨
a
,
b

|


a

2


b

a

2



b

−
1


,

b

2


a

b

2



a

−
1


⟩


{\displaystyle \langle a,b|a^{2}ba^{2}b^{-1},b^{2}ab^{2}a^{-1}\rangle }

is torsion-free but not left-orderable; 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 unit conjecture. More generally the topic of orderability of 3--manifold groups is interesting for its relation with various topological invariants. There exists a 3-manifold group which is left-orderable but not bi-orderable (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 systems as it is known that a countable group is left-orderable if and only if it acts on the real line by homeomorphisms. Non-examples related to this paradigm are lattices in higher rank Lie groups; it is known that (for example) finite-index subgroups in





S
L


n


(

Z

)


{\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )}

are not left-orderable; a wide generalisation of this has been recently announced.




See also


Cyclically ordered group
Hahn embedding theorem
Partially ordered group




Notes






References


Deroin, Bertrand; Navas, Andrés; Rivas, Cristóbal (2014). "Groups, orders and dynamics". arXiv:1408.5805 [math.GT].
Levi, F.W. (1942), "Ordered groups.", Proc. Indian Acad. Sci., A16 (4): 256–263, doi:10.1007/BF03174799, S2CID 198139979
Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715
Ghys, É. (2001), "Groups acting on the circle.", L'Enseignement Mathématique, 47: 329–407

Kata Kunci Pencarian:

• Linearly ordered group
• Partially ordered group
• Archimedean property
• Archimedean group
• Ordered field
• Total order
• Cyclic order
• Ordered ring
• Hahn embedding theorem
• Cyclically ordered group