Talk:Max Operation Representation on Real Numbers

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Undefined entities?

The language in which this is couched seems to in the t.o.a.g. down to one in which addition, subtraction and the modulus operation are all defined. As far as the definition goes, those operations so far are defined only within the context of conventional arithmetic.

It is of course trivial to define the concepts of subtraction (the inverse of the group operation addition), but defining the modulus / absolute value is not quite as straightforward and needs to be done with care.

I also take issue with the fact that $\frac 1 2$ is an operation even more specific to the idea of numbers: for a start, $2$ is an object which is not necessarily an element of a general t.o.a.g. in the first place, let alone the concept of dividing $1$ by it.

I also encourage the use of a general "non-specific" operator like $\circ$ rather than $+$. The latter is too easily conflated with number addition, and this theorem applies to a deliberately more general class of objects than just sets of numbers.

Consequently, if this page is to survive at all, it needs to look more like:

$\max \left\{{x,y}\right\} \circ \max \left\{{x,y}\right\} = x \circ y \circ \left\vert{x \circ \left({y^{-1}}\right)}\right\vert$

where $\left\vert{x \circ \left({y^{-1}}\right)}\right\vert$ is defined as:

$\forall a, b \in G: \left\vert{a \circ b}\right\vert := \begin{cases} a \circ b & : b \preceq a \\ b \circ a & : a \preceq b \end{cases}$

... although you would put that last definition in a separate page and define it in the context of a t.o.a.g.

I recommend you go back to your source work to investigate the context into which this theorem is introduced, because as it stands it is lacking for the reasons stated above.


As for your final comment "Show me how you would clean this up", the first thing I will note is the line which you have started "Where" should be "where". Think about what you are saying linguistically. You are writing a sentence: "This interesting thing holds for $x$, where $x$ means this". The fact that the statement includes a mathematical expression should not confuse the eye into thinking that just because the "where" goes on the next line it starts a new sentence.

As for the rest of it, investigate the differences before and after the changes (if and when the above comments are either acted on or commented on further), so as to see what as changed as a result of the tidy-up. --prime mover (talk) 14:50, 21 December 2012 (UTC)

I originally came up with this working with $\R$ because the max function seemed a little bit unnatural to me. I tried to generalise it to a totally ordered integral domain (for which this wiki has the absolute value function defined) but could not see how $\times$ was relevant. So I dropped it and worked with a t.o.a.g.
I agree to feeling uncomfortable about using $\frac 1 2$ and $+$ and merely did so for readability. Although I feel it could be generalised I will narrow it down to $\R$ and not add to ProofWiki's burdens. --Jshflynn (talk) 15:27, 21 December 2012 (UTC)
I think there's definitely mileage in your idea - just that more background work needs to be done. --prime mover (talk) 15:49, 21 December 2012 (UTC)