Wikipedia / Linearly ordered group
Wikipedia / 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
with identity element being right-invariant. In particular, a group being left-orderable is the same as it being right-orderable.
of an ordered group positive. The set of positive elements in an ordered group is called the positive cone is used for the positive cone together with the identity element.
satisfying the two conditions that:
;
is the disjoint union.
.
as well. This is equivalent to the positive cone being stable under inner automorphisms.
, then the absolute value is abelian a triangle inequality.
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, . If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completionth roots. We endow this space with the usual topology 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.
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 are not left-orderable; a wide generalisation of this was announced in 2020.
See also
Notes