Let $G$ be a group and $\leq$ be a partial order. $G$ is partially ordered by $\leq$ if for any $a$, $b$, $x\in G$,
\[a\leq b \enspace\rimp\enspace ax\leq bx \,\land\, xa\leq xb.\]$G$ is totally ordered if $\leq$ is total.
Let $G$ be a group and $\leq$ be a partial order. $G$ is partially ordered by $\leq$ if for any $a$, $b$, $x\in G$,
\[a\leq b \enspace\rimp\enspace ax\leq bx \,\land\, xa\leq xb.\]$G$ is totally ordered if $\leq$ is total.