Thread Rating:
  • 1 Vote(s) - 4 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Negative, Fractional, and Complex Hyperoperations
Is there a way to continue the patterns we see within the natural numbers of current hyper-operations (Hyper-1, Hyper-2, Hyper-3, Hyper-4, ect...) or at least prove that we cannot extend the value of the operation to fractional numbers? E.g. Hyper-1/2. Negative numbers? E.g. Hyper-(-2) Or even imaginary numbers? E.g. Hyper-3i.
They need not be defined, but are these operations technically there, just without practical use? Or are our names for the hyper-operations strictly for listing and naming purposes, with no way to derive meaning from such a number?
Could a fractional, or negative hyper-operation represent an operator we have already defined? E.g. Hyper-(-2)= Division, or Hyper-1/2 = Division?
Comments on the controversy of Zeration are also encouraged.
-rank hyperoperations have meaning as long as we can iterate times a function defined in the set of the binary functions over the naturals numbers (or defined over a set of binary functions.)

let me explain why.

There are many differente Hyperoperations sequences, end they are all defined in a different way:

we start with an operation and we obtain its successor operation applying a procedure (usually a recursive one).

So every Hyperoperation sequence is obtained applying that recursive procedure to a base operation (aka the first step of the sequence)

and so on

or in a formal way

That is the same as

so if we can extend the iteration of from to the real-complex numbers the work is done.


MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
I'm not sure but I think that bo198214(Henrik Trappmann) had this idea in 2008

With his idea we can reduce the problem of real-rank hyperoperations to an iteration problem

Later this idea was better developed by JmsNxn (2011) with the concept of "meta-superfunctions"

I'm still working on his point of view but there is a lot of work to do...

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)

Possibly Related Threads…
Thread Author Replies Views Last Post
  A relaxed zeta-extensions of the Recursive Hyperoperations MphLee 3 3,823 06/06/2022, 07:37 PM
Last Post: MphLee
  On my old fractional calculus approach to hyper-operations JmsNxn 14 6,135 07/07/2021, 07:35 AM
Last Post: JmsNxn
  @Andydude References about the formalization of the Hyperoperations MphLee 3 7,828 07/25/2014, 10:41 AM
Last Post: MphLee
  Theorem in fractional calculus needed for hyperoperators JmsNxn 5 13,037 07/07/2014, 06:47 PM
Last Post: MphLee
  Easy tutorial on hyperoperations and noptiles MikeSmith 2 5,947 06/26/2014, 11:58 PM
Last Post: MikeSmith
  left-right iteraton in right-divisible magmas, and fractional ranks. MphLee 1 5,395 05/14/2014, 03:51 PM
Last Post: MphLee
  A new way of approaching fractional hyper operators JmsNxn 0 5,698 05/26/2012, 06:34 PM
Last Post: JmsNxn
  generalizing the problem of fractional analytic Ackermann functions JmsNxn 17 39,561 11/24/2011, 01:18 AM
Last Post: JmsNxn
  Operations with fractional index between + and * ? Gottfried 6 15,407 10/21/2009, 01:30 AM
Last Post: andydude

Users browsing this thread: 1 Guest(s)