In

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, specifically abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

, a linearly ordered or totally ordered group is a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

''G'' equipped with a total order
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

"≤" 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 exists left-orderable groups which are not bi-orderable.
Further definitions

In this section $\backslash le$ is a left-invariant order on a group $G$ withidentity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

$e$. All that is said applies to right-invariant orders with the obvious modifications. Note that $\backslash le$ being left-invariant is equivalent to the order $\backslash le\text{'}$ defined by $g\; \backslash le\text{'}\; h$ if and only if $h^\; \backslash le\; g^$ 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 $g\; \backslash not=\; e$ of an ordered group positive if $e\; \backslash le\; g$. The set of positive elements in an ordered group is called the positive cone, it is often denoted with $G\_+$; the slightly different notation $G^+$ is used for the positive cone together with the identity element.
The positive cone $G\_+$ characterises the order $\backslash le$; indeed, by left-invariance we see that $g\; \backslash le\; h$ if and only if $g^\; h\; \backslash in\; G\_+$. In fact a left-ordered group can be defined as a group $G$ together with a subset $P$ satisfying the two conditions that:
#for $g,\; h\; \backslash in\; P$ we have also $gh\; \backslash in\; P$;
#let $P^\; =\; \backslash $, then $G$ is the disjoint union
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of $P,\; P^$ and $\backslash $.
The order $\backslash le\_P$ associated with $P$ is defined by $g\; \backslash le\_P\; h\; \backslash Leftrightarrow\; g^\; h\; \backslash in\; P$; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of $\backslash le\_P$ is $P$.
The left-invariant order $\backslash le$ is bi-invariant if and only if it is conjugacy invariant, that is if $g\; \backslash le\; h$ then for any $x\; \backslash in\; G$ we have $xgx^\; \backslash le\; xhx^$ as well. This is equivalent to the positive cone being stable under inner automorphism
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s.
If $a\; \backslash in\; G$, then the absolute value of $a$, denoted by $,\; a,$, is defined to be: $$,\; a,\; :=\backslash begina,\; \&\; \backslash texta\; \backslash ge\; 0,\backslash \backslash \; -a,\; \&\; \backslash text.\backslash end$$
If in addition the group $G$ is abelian, then for any $a,\; b\; \backslash in\; G$ a triangle inequality
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

is satisfied: $,\; a+b,\; \backslash le\; ,\; a,\; +,\; b,$.
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-freeabelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

is bi-orderable; this is still true for nilpotent group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s but there exists torsion-free, finitely presented group
In mathematics, a presentation is one method of specifying a group (mathematics), group. A presentation of a group ''G'' comprises a set ''S'' of generating set of a group, generators—so that every element of the group can be written as a produ ...

s which are not left-orderable.
Archimedean ordered groups

Otto Hölder showed that everyArchimedean group In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...

(a bi-ordered group satisfying an Archimedean property
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

) is isomorphic
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

to a subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...

of the additive group of real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s, .
If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion, $\backslash widehat$ of the closure of a l.o. group under $n$th roots. We endow this space with the usual topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of a linear order, and then it can be shown that for each $g\backslash in\backslash widehat$ the exponential maps $g^:(\backslash mathbb,+)\backslash to(\backslash widehat,\backslash cdot)\; :\backslash lim\_q\_\backslash in\backslash mathbb\backslash mapsto\; \backslash lim\_g^$ are well defined order preserving/reversing, topological group
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...

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 group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s are left-orderable. More generally this is also the case for right-angled Artin group
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two rays lie in the plane (g ...

s. Braid group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s are also left-orderable.
The group given by the presentation $\backslash langle\; a,\; b\; ,\; a^2ba^2b^,\; b^2ab^2a^\backslash rangle$ is torsion-free but not left-orderable; note that it is a 3-dimensional crystallographic group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchan ...

(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 system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the ...

s cas 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 $\backslash mathrm\_n(\backslash mathbb\; Z)$ are not left-orderable; a wide generalisation of this has been recently announced.
See also

*Cyclically ordered group
In mathematics, a cyclically ordered group is a Set (mathematics), set with both a group (mathematics), group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order.
Cyclically ordered groups were firs ...

*Hahn embedding theorem
In mathematics – especially in the area of abstract algebra dealing with ordered structures on abelian groups – the Hahn embedding theorem gives a simple description of all linearly ordered group, linearly ordered abelian groups. It is named a ...

*Partially ordered group
In abstract algebra, a partially ordered group is a group (mathematics), group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', ...

Notes

References

* * * *{{Citation , last1=Ghys , first1=É. , title=Groups acting on the circle. , journal=L'Enseignement Mathématique , year=2001 , volume=47 , pages=329–407 Ordered groups