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Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb From Wikipedia, the free encyclopedia

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.

This section includes a list of general references, but it lacks sufficient corresponding inline citations. (July 2024) |

In this section is a left-invariant order on a group with identity element . All that is said applies to right-invariant orders with the obvious modifications. Note that being left-invariant is equivalent to the order defined by if and only if 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 of an ordered group **positive** if . The set of positive elements in an ordered group is called the **positive cone**, it is often denoted with ; the slightly different notation is used for the positive cone together with the identity element.^{[1]}

The positive cone characterises the order ; indeed, by left-invariance we see that if and only if . In fact a left-ordered group can be defined as a group together with a subset satisfying the two conditions that:

- for we have also ;
- let , then is the disjoint union of and .

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

The left-invariant order is bi-invariant if and only if it is conjugacy invariant, that is if then for any we have as well. This is equivalent to the positive cone being stable under inner automorphisms.

If ^{[citation needed]}, then the **absolute value** of , denoted by , is defined to be:
If in addition the group is abelian, then for any a triangle inequality is satisfied: .

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;^{[2]} this is still true for nilpotent groups^{[3]} but there exist torsion-free, finitely presented groups which are not left-orderable.

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, of the closure of a l.o. group under th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each the exponential maps 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.

Free groups are left-orderable. More generally this is also the case for right-angled Artin groups.^{[4]} Braid groups are also left-orderable.^{[5]}

The group given by the presentation is torsion-free but not left-orderable;^{[6]} 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.^{[7]} There exists a 3-manifold group which is left-orderable but not bi-orderable^{[8]} (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.^{[9]} Non-examples related to this paradigm are lattices in higher rank Lie groups; it is known that (for example) finite-index subgroups in are not left-orderable;^{[10]} a wide generalisation of this has been recently announced.^{[11]}

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