\begin{definition}[Definition:Archimedean Property/Ordering]
Let $\struct {S, \circ}$ be a closed algebraic structure.
Let $\cdot: \Z_{>0} \times S \to S$ be the operation defined as:
:$m \cdot a = \begin{cases}
a & : m = 1 \\
a \circ \paren {\paren {m - 1} \cdot a} & : m > 1 \end {cases}$
Let $\preceq$ be an ordering on $S$.
Then $\preceq$ satisfies the '''Archimedean property on $S$''' {{iff}}:
:$\forall a, b \in S: a \prec b \implies \exists m \in \Z_{>0}: b \prec m \cdot a$
Category:Definitions/Abstract Algebra
Category:Definitions/Order Theory
\end{definition}