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Generalized arithmetic operator
#16
Superfunction is a multivalued function defined over a set of functions not over a set of numbers:

means that takes a function and gives a function calles superfunction of

such that satisfies

1)

since there are infinite solution for (infinite superfunctions) that means that is multivalued and then is not a function at all and we have to put some restrictions:
using Trapmann-Kouznetsov terminology used in their paper "5+ methods..." we call the -based superfunction of the function that satifies two requirements

1)

2)


and we have



In this way we obtain uniqueness over the naturals: in fact superfunction is equivalent to the "definition by recursion" that is unique .
But is not over the reals... there we need more requirments.

Obviously this is still not enough to achieve the uniqueness of (iteration of ) that would mean having to be a function over a set of functions (not multivalued).

By the way I guess that Trapmann and Kouznetsov tried to find such additionals requirments but my math level is not enough to understand it.

Anyways we have that is a function and is the half superfunction.

example :
let define and we have

(multiplication is the 0-based superfunction of addition)

so we search for a such that



and that if we should have (maybe...)


and




I apologize if I did some mistakes.
MathStackExchange account:MphLee
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Messages In This Thread
Generalized arithmetic operator - by hixidom - 03/11/2014, 03:52 AM
RE: Generalized arithmetic operator - by JmsNxn - 03/11/2014, 03:15 PM
RE: Generalized arithmetic operator - by hixidom - 03/11/2014, 06:24 PM
RE: Generalized arithmetic operator - by MphLee - 03/11/2014, 10:49 PM
RE: Generalized arithmetic operator - by hixidom - 03/11/2014, 11:20 PM
RE: Generalized arithmetic operator - by MphLee - 03/12/2014, 11:18 AM
RE: Generalized arithmetic operator - by JmsNxn - 03/12/2014, 02:59 AM
RE: Generalized arithmetic operator - by hixidom - 03/12/2014, 04:37 AM
RE: Generalized arithmetic operator - by MphLee - 03/12/2014, 06:19 PM
RE: Generalized arithmetic operator - by hixidom - 03/12/2014, 06:43 PM
RE: Generalized arithmetic operator - by hixidom - 03/22/2014, 12:06 AM
RE: Generalized arithmetic operator - by hixidom - 03/22/2014, 12:42 AM
RE: Generalized arithmetic operator - by hixidom - 06/11/2014, 05:10 PM

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