Definition:Archimedean Property

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Definition

Let $\struct {S, \circ}$ 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:

$m \cdot a = \begin{cases} a & : m = 1 \\ a \circ \paren {\paren {m - 1} \cdot a} & : m > 1 \end {cases}$

Archimedean Property on Norm

Let $n: S \to \R$ be a norm on $S$.

Then $n$ satisfies the Archimedean property on $S$ if and only if:

$\forall a, b \in S: n \paren a < n \paren b \implies \exists m \in \N: n \paren {m \cdot a} > n \paren b$

Using the more common symbology for a norm:

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

Archimedean Property on Ordering

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

Then $\preceq$ satisfies the Archimedean property on $S$ if and only if:

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

Source of Name

This entry was named for Archimedes of Syracuse.