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at triangle inequality the discussion of the interpretation in enriched category theory had been missing. I have added in a corrresponding section here and cross-linked with Lawvere metric spaces.
Odd that a condition is named $X$ and then a stronger condition is named ’non-$X$’, as with archimedean. It’s not as though the latter does not satisfy the former condition too, so why ’non’?
the unitality condition is part of the non-degeneracy condition on a me,
I can’t even guess the correction.
I’m guessing “me” should be “metric”, and the non-degeneracy condition is $d(x, x) = 0$ iff $x = 0$, where unitality gives only the “if” half.
Re #2, I guess the Archimedean property explains things.
Re #4
$d(x, y) = 0$ iff $x = y$?
I added in at absolute value an explanation for ’non-archimedean’:
If the last triangle inequality is strengthened to
- ${\vert x + y \vert} \leq max({\vert x \vert}, {\vert y \vert})$
then ${\vert {-} \vert}$ is called an ultrametric or non-archimedean absolute value, since then for any $x, y \in k$ with $\vert x \vert \lt \vert y \vert$ then for all natural numbers $n$, $\vert n x \vert \leq \vert x \vert \lt \vert y \vert$. Otherwise it is called archimedean.
Sorry for that. Thanks, Todd!, and thanks David for fixing the entry.
I don’t think I’ve ever heard an ultrametric (not arising from a norm) called “non-archimedean”, although Wikipedia mentions it as an alternative terminology. I agree it seems wrong for a stronger property to get a “non-” name. I certainly haven’t heard a not-necessarily-ultra-metric called “archimedean”. I would suggest we reserve that terminology for the normed case.
Sorry, did I write this? I forget what happened here. Let’s fix this.
I changed absolute value to say that the archimedean ones are those which actually satisfy the archimedean property. Is every absolute value either archimedean (in this sense) or non-archimedean (in that the ultrametric property holds)?
I don't think that anybody is suggesting that one call a non-necessarily-ultra-metric ‘archimedean’. So neither notion is stronger than the other; they are are mutually exclusive.
But I don't really understand this stuff and would also appreciate an answer to Mike's question: is this equivalent to the archimedean property from algebra, and how? At Wikipedia/EN:Archimedean property#Definition for normed fields, it is stated to be true for absolute values on fields (in which case the algebraic property is $\forall\, x\colon K,\; \exists\, n\colon \mathbb{N},\; {|x/n|} \lt 1$), but there is no reason given or reference cited.
We should surely have a separate page on the Archimedean property/axiom. The EOM has Archimedean axiom:
An axiom, originally formulated for segments, which states that if the smaller one of two given segments is marked off a sufficient number of times, it will always produce a segment larger than the larger one of the original two segments. This axiom can be formulated in an analogous manner for surfaces, volumes, positive numbers, etc. In general, the Archimedean axiom applies to a given quantity if for any two values $A$ and $B$ of this quantity such that $A \lt B$ it is always possible to find an integer $m$ such that $A m\gt B$.
Presumably that’s the basic idea that positive quantities are comparable. But then one needs to speak of situations with inverses such as Archimedean group.
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