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 ${\displaystyle G}$ with identity element ${\displaystyle e}$. 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 ${\displaystyle g\leq 'h}$ if and only if ${\displaystyle h^{-1}\leq g^{-1}}$ 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 ${\displaystyle g\not =e}$ of an ordered group positive if ${\displaystyle e\leq g}$. The set of positive elements in an ordered group is called the positive cone, it is often denoted with ${\displaystyle G_{+}}$; the slightly different notation ${\displaystyle G^{+}}$ is used for the positive cone together with the identity element.^[1]

The positive cone ${\displaystyle G_{+}}$ characterises the order ${\displaystyle \leq }$; indeed, by left-invariance we see that ${\displaystyle g\leq h}$ if and only if ${\displaystyle g^{-1}h\in G_{+}}$. In fact, a left-ordered group can be defined as a group ${\displaystyle G}$ together with a subset ${\displaystyle P}$ satisfying the two conditions that:

1. for ${\displaystyle g,h\in P}$ we have also ${\displaystyle gh\in P}$;
2. let ${\displaystyle P^{-1}=\{g^{-1}\mid g\in P\}}$, then ${\displaystyle G}$ is the disjoint union of ${\displaystyle P,P^{-1}}$ and ${\displaystyle \{e\}}$.

The order ${\displaystyle \leq _{P}}$ associated with ${\displaystyle P}$ is defined by ${\displaystyle g\leq _{P}h\Leftrightarrow g^{-1}h\in P}$; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of ${\displaystyle \leq _{P}}$ is ${\displaystyle P}$.

The left-invariant order ${\displaystyle \leq }$ is bi-invariant if and only if it is conjugacy-invariant, that is if ${\displaystyle g\leq h}$ then for any ${\displaystyle x\in G}$ we have ${\displaystyle xgx^{-1}\leq xhx^{-1}}$ as well. This is equivalent to the positive cone being stable under inner automorphisms.

If ${\displaystyle a\in G}$^{[citation needed]}, then the absolute value of ${\displaystyle a}$, denoted by ${\displaystyle |a|}$, is defined to be: $${\displaystyle |a|:={\begin{cases}a,&{\text{if }}a\geq 0,\\a^{-1},&{\text{otherwise}}.\end{cases}}}$$ If in addition the group ${\displaystyle G}$ is abelian, then for any ${\displaystyle a,b\in G}$ a triangle inequality is satisfied: ${\displaystyle |a+b|\leq |a|+|b|}$.

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;^[2] this is still true for nilpotent groups^[3] 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, ${\displaystyle {\widehat {G}}}$ of the closure of a l.o. group under ${\displaystyle n}$th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each ${\displaystyle g\in {\widehat {G}}}$ the exponential maps ${\displaystyle g^{\cdot }:(\mathbb {R} ,+)\to ({\widehat {G}},\cdot ):\lim _{i}q_{i}\in \mathbb {Q} \mapsto \lim _{i}g^{q_{i}}}$ 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.^[4] Braid groups are also left-orderable.^[5]

The group given by the presentation ${\displaystyle \langle a,b|a^{2}ba^{2}b^{-1},b^{2}ab^{2}a^{-1}\rangle }$ 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 ${\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )}$ are not left-orderable;^[10] a wide generalisation of this was announced in 2020.^[11]

See also

• Cyclically ordered group
• Hahn embedding theorem
• Partially ordered group