Definition talk:Archimedean Property/Norm

I could see nothing wrong with the original definition as given:
 * 1) The definition of $\cdot$ is a generalisation of the "power" operation to a not-necessarily-associative algebraic structure. The association is specifically defined as right-to-left, although this could be defined either way. The operation does not need to be defined on a semigroup.
 * 2) The definition of the property:
 * $\forall a, b \in S: n \left({a}\right) < n \left({b}\right) \implies \exists m \in \N: n \left({m \cdot a}\right) > n \left({b}\right)$

is perfectly valid, from what I can tell. If the norm of $a$ is less than the norm of $b$, then the norm of some finite power of $a$ (as defined above) will be greater than $b$. It effectively says that for sufficiently high $m$, the norm of the power of any element will exceed the norm of any other given element.

In your definition, you specify that "if the norm of $a$ is greater than zero" - I am led to understand that this is one of the criteria for a norm to be a norm, so that goes without saying.

Consequently I have reverted your changes.

Feel free to discuss why you believe the original page was wrong. --prime mover (talk) 07:21, 28 October 2012 (UTC)