Definition:Archimedean Property/Ordering

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

Let $\cdot: \N_{>0} \times S \to S$ be defined as:
 * $\forall a \in S: \forall m \in \N_{>0}: m \cdot a = \begin{cases}

a & : m = 1 \\ a \circ \left({\left({m - 1}\right) \cdot a}\right) & : m > 1 \end {cases}$

Let $\left({S, \circ \preceq}\right)$ be an ordered monoid.

Let $e$ be the identity element of $\left({S, \circ}\right)$.

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


 * $\forall a, b \in S: e \prec a \implies \exists m \in \N_{>0}: b \prec m \cdot a$

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