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Octonion Hyper-Operations
#1
Question 
How does tetration work for quaternions? What about for octonions? How do pentation and higher hyper-operations work for those?
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ
Please remember to stay hydrated.
Sincerely: Catullus
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#2
We try to walk before we can run Smile

We do not even understand tetration well for octonions ( or even complex numbers lol )

And there are many tetrations and many hyper-operators.

So I assume it might be case specific.

I like the question , but it seems very very general.

Wish I could say more ... sorry.


regards

tommy1729
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#3
I think most (known) methods rely on properties such as commutative associative etc.

So that might imply they all fail for most non-complex numbers.

Note that it is also problematic to define derivatives for non-complex functions. ( zero-divisors , depending on the type of infinitesimal , ... ) 
so that makes formulas that use calculus concepts * uncomfortable *. 

Functional composition is associative ... but if we plug in non-associative numbers as parameters ... feels * delicate *.

Also we might get extra fixpoints in say the quaternions and such that harrass our convergeance domains and such.
Or as we learned before ; functions do no agree on the fixpoint. ( see the bummer thread )

There are also issues with combinatorial interpretations/methods but those are complicated.

Best I hope for is that the taylor series expansions of the solutions we have can be used ;

plug in a power-associative number x instead of the complex z and just compute the values.
... It should agree with the complex solution and by extension also the field/ring/whatever of x.

hopefully.



regards

tommy1729
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#4
Question 
How do they work for matrices?
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ
Please remember to stay hydrated.
Sincerely: Catullus
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