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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 \le is a left-invariant order on a group G with
identity 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 \le being left-invariant is equivalent to the order \le' defined by g \le' h if and only if h^ \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 \not= e of an ordered group positive if e \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 \le; indeed, by left-invariance we see that g \le h if and only if g^ h \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 \in P we have also gh \in P; #let P^ = \, 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 ...

disjoint union
of P, P^ and \. The order \le_P associated with P is defined by g \le_P h \Leftrightarrow g^ 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 \le_P is P. The left-invariant order \le is bi-invariant if and only if it is conjugacy invariant, that is if g \le h then for any x \in G we have xgx^ \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 \in G, then the absolute value of a, denoted by , a, , is defined to be: , a, :=\begina, & \texta \ge 0,\\ -a, & \text.\end If in addition the group G is abelian, then for any a, b \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 ...

triangle inequality
is satisfied: , a+b, \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-free
abelian 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 every
Archimedean 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), ...

Archimedean property
) 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 ...

isomorphic
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, \widehat of the closure of a l.o. group under nth 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 ...

topology
of a linear order, and then it can be shown that for each g\in\widehat the exponential maps g^:(\mathbb,+)\to(\widehat,\cdot) :\lim_q_\in\mathbb\mapsto \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 \langle a, b , a^2ba^2b^, b^2ab^2a^\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 \mathrm_n(\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