Definition:Archimedean Property/Ordering

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

Let $\cdot: \Z_{>0} \times S \to S$ be the operation defined as:
 * $m \cdot a = \begin{cases}

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

Let $\preceq$ be an ordering on $S$.

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


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