\begin{definition}[Definition:Archimedean Property/Norm]
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 $n: S \to \R$ be a norm on $S$.
{{explain|What is a norm on a general algebraic structure?}}
{{Disambiguate|Definition:Norm}}
Then $n$ satisfies the '''Archimedean property on $S$''' {{iff}}:
:$\forall a, b \in S: n \paren a < n \paren b \implies \exists m \in \N: n \paren {m \cdot a} > n \paren b$
Using the more common symbology for a norm:
:$\forall a, b \in S: \norm a < \norm b \implies \exists m \in \Z_{>0}: \norm {m \cdot a} > \norm b$
Category:Definitions/Abstract Algebra
Category:Definitions/Norm Theory
\end{definition}