Definition:Archimedean Property
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Contents
Definition
Let $\left({S, \circ}\right)$ be a closed algebraic structure on which there exists either an ordering or a norm.
Let $\cdot: \Z_{>0} \times S \to S$ be the operation defined as:
- $m \cdot a = \begin{cases} a & : m = 1 \\ a \circ \left({\left({m - 1}\right) \cdot a}\right) & : m > 1 \end {cases}$
Archimedean Property on Norm
Let $n: S \to \R$ be a norm on $S$.
What is a norm on a general algebraic structure?
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Then $n$ satisfies the Archimedean property on $S$ iff:
- $\forall a, b \in S: n \left({a}\right) < n \left({b}\right) \implies \exists m \in \N: n \left({m \cdot a}\right) > n \left({b}\right)$
Using the more common symbology for a norm:
- $\forall a, b \in S: \left \Vert{a}\right \Vert < \left \Vert{b}\right \Vert \implies \exists m \in \Z_{>0}: \left \Vert{m \cdot a}\right \Vert > \left \Vert{b}\right \Vert$
Archimedean Property on Ordering
Let $\preceq$ be an ordering on $S$.
Then $\preceq$ satisfies the Archimedean property on $S$ iff:
- $\forall a, b \in S: a \prec b \implies \exists m \in \Z_{>0}: b \prec m \cdot a$
Source of Name
This entry was named for Archimedes of Syracuse.
