# Definition:Archimedean Property/Ordering

## Definition

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 $S$ if and only if:

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