Definition:Archimedean Property/Ordering

Definition
Let $\left({S, \circ}\right)$ be a semigroup.

Let $\left({S, \circ \preceq}\right)$ be a totally ordered semigroup.

Then $\preceq$ satisfies the Archimedean property on $S$ iff:


 * $\forall a, b \in S: a \prec b \implies \exists m \in \N_{>0}: \left({b \prec a^m}\right) \lor \left({b^m \prec a}\right)$

where $x^m$ denotes the $m$th power of $x$.

We say that $\left({S, \circ \preceq}\right)$ is an Archimedean totally ordered semigroup.