linearly ordered group

A linearly ordered group is a group $G$ equipped with a total order $\le$ ; specifically, it is a

• left-ordered group if $\le$ is left-invariant, i.e. $a\le b⟹ca\le cb$ for all $a,b,c\in G$,
• right-ordered group if $\le$ is right-invariant, i.e. $a\le b⟹ac\le bc$ for all $a,b,c\in G$,
• bi-ordered group if $\le$ is both left-invariant and right-invariant.