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I asked a question yesterday regarding numbers defined on graph structures that I call graph numbers. I posted the algorithm I was using to define graph numbers, which are simply a natural extension of linear numbers we use all the time. Nolion's response was close but not quite right. The equations are actually:

$d_{m+n}(v) = \biggl (d_{n}(v) + d_{m}(v) + \sum_{(u,v)\in{E}}{d_{{m+n},{carry over}}(u)} \biggr ) \mod {2}$

$d_{{m+n},carryover}(v) = \biggl \lfloor \frac {d_{n}(v) + d_{m}(v) + \sum_{(u,v)\in{E}}{d_{{m+n},{carry over}}(u)} } {2} \biggr \rfloor$

After reading nolion's response, I was able to come up with these equations. Notice, they are added in $Z$, which is why I need the $\mod 2$ to bring them back to $Z/2Z$. However, the digits of m and n are chosen from $Z/2Z$ (just as in normal binary addition of more than 2 numbers). But the order of evaluation of $d_{m+n}$ matters (which, again, I am not sure how to describe mathematically) starting from the "root" nodes and evaluating as in a Breadth First Search. Notice, that if the Directed Acyclic Graph is a linear connection of nodes, this becomes normal arithmetic in base 2. My question is, has there been any work done on such numbers?

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    @Magidin: I guess we'll just have to agree to disagree on the "usefulness" issue. It's more of a philosophical point anyways.2011-02-19

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I had a nice, long answer written out that has been subsumed by all the comments above. Oh well.

The only part that isn't redundant is that for multiplication, you want to look at whether the set of endomorphisms of your DAG acts vertex transitively. Ideally, it would act simply vertex-transitively, because otherwise, you would have to include choices of endomorphisms explicitly in the structure. Anyway, that's the mathematical terminology for your "fractal structure".

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    @Aubrey da Cunha: Thanks. I will definitely show this to my professor.2011-02-20