A relaxed -extensions of the Recursive Hyperoperations

I want to show you an easy extension for hyperoperations.

I don't want it to be the most natural, but I want to ask if someone already used this extension and if it can be usefull for something.

Since is a bit different I want to use the plus-notation () for the hyperoperators.

I start with these basic definitions over the naturals :

\( o')\,\,\,B_b( \sigma+1):=

\begin{cases}

b, & \text{if} \sigma=0 \\

0, & \text{if} \sigma=1 \\

1, & \text{if} \sigma\gt 1 \\

\end{cases}\)

Then the recursive definitions of the operators

Observation before the extension's definitons

we can see that from rank zero to rank one we can define infinite functions with

Generalizing, now we can define as a continous functions from the interval to , to the interval to :

\( Ev)\,\,\,\zeta_b(\varepsilon)=\begin{cases}

1, & \text{if $\varepsilon=0$} \\

b, & \text{if $\varepsilon=1$ } \\ \end{cases} \)

And we can define the operations with fractional rank starting from the interval

\( Evi)\,\,\, b +_{\varepsilon}n=\zeta _b(\varepsilon)+_{1}n \,\, \text{ and} \,\, \varepsilon \in [0,1] \)

Other operations are these ( and ):

Example of functions and the generated -hyperoperations:

and

for and

I want to show you an easy extension for hyperoperations.

I don't want it to be the most natural, but I want to ask if someone already used this extension and if it can be usefull for something.

Since is a bit different I want to use the plus-notation () for the hyperoperators.

I start with these basic definitions over the naturals :

\( o')\,\,\,B_b( \sigma+1):=

\begin{cases}

b, & \text{if} \sigma=0 \\

0, & \text{if} \sigma=1 \\

1, & \text{if} \sigma\gt 1 \\

\end{cases}\)

Then the recursive definitions of the operators

Observation before the extension's definitons

we can see that from rank zero to rank one we can define infinite functions with

Generalizing, now we can define as a continous functions from the interval to , to the interval to :

\( Ev)\,\,\,\zeta_b(\varepsilon)=\begin{cases}

1, & \text{if $\varepsilon=0$} \\

b, & \text{if $\varepsilon=1$ } \\ \end{cases} \)

And we can define the operations with fractional rank starting from the interval

\( Evi)\,\,\, b +_{\varepsilon}n=\zeta _b(\varepsilon)+_{1}n \,\, \text{ and} \,\, \varepsilon \in [0,1] \)

Other operations are these ( and ):

Example of functions and the generated -hyperoperations:

and

for and

MSE MphLee

Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)

S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)