Linearly ordered group

• 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 ${\displaystyle \le }$ 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 ${\displaystyle \le }$ being left-invariant is equivalent to the order ${\displaystyle {\le }^{\prime }}$ defined by ${\displaystyle g{\le }^{\prime }h}$ if and only if ${\displaystyle h^{-1}\le g^{-1}}$ 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 ${\displaystyle g\ne e}$ of an ordered group positive if ${\displaystyle e\le g}$. 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.

The positive cone ${\displaystyle G_{+}}$ characterises the order ${\displaystyle \le }$; indeed, by left-invariance we see that ${\displaystyle g\le h}$ 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 ${\displaystyle P^{-1}=\{g^{-1},g\in P\}}$, then ${\displaystyle G}$ is the disjoint union of ${\displaystyle P,P^{-1}}$ and ${\displaystyle \{e\}}$.

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

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

If ${\displaystyle a\in G}$, then the absolute value of ${\displaystyle a}$, denoted by ${\displaystyle |a|}$, is defined to be: $${\displaystyle |a|:=\{\begin{array}{ll}a,& \text{if }a\ge 0,\\ -a,& \text{otherwise}.\end{array}}$$ If in addition the group ${\displaystyle G}$ is abelian, then for any ${\displaystyle a,b\in G}$ a triangle inequality is satisfied: ${\displaystyle |a+b|\le |a|+|b|}$.