Hyper-operator Terminology

Hyper-operators are an increasingly general class of binary operations that include addition, multiplication, and exponentiation. There are many different kinds of hyper-operators that have successfully spanned these three operations, but unless otherwise specified, usually the first kind are implied.

The different kinds of hyper-operators discussed on this site are:

Hyper-operator Notation

Higher Hyper-operators

Mixed Hyper-operators

Lower Hyper-operators

Frappier Hyper-operators

Frappier hyper-operators are defined by the relation:

  H\!F_n(x, y) = (x \mapsto H\!F_{n-1}(x, x))^{\log_2(y)}(x)

This definition has the following consequences:

    H\!F_1(x, y) & = x + y \\\notag
    H\!F_2(x, y) & = x y \\\notag
    H\!F_3(x, y) & = x^y \\\notag
    H\!F_4(x, y) & = (x \mapsto x^x)^{\log_2(y)}(x)

It can be shown that the zeroth Frappier hyper-operator cannot exist, because 
  x + 1 = H\!F_1(x, 1) = (x \mapsto H\!F_0(x, x))^{0}(x) = x
  by definition. Since this is a contradiction, we cannot have a zeroth Frappier hyper-operator.

Robbins Hyper-operators

Robbins hyper-operators are defined by the relation:

  H\!R_n(x, y) = \exp_b H\!R_{n-1}\left( \log_b(x), \exp_b^{n-3}(y) \right)

This definition has the following consequences (for b = e):

    H\!R_{(-1)}(x, y) & = \log^2\left(\exp^7(y) + e^{e^x}\right) \\\notag
    H\!R_0(x, y) & = \log\left(e^{e^{e^y}} + e^x\right) \\\notag
    H\!R_1(x, y) & = x + y \\\notag
    H\!R_2(x, y) & = x y \\\notag
    H\!R_3(x, y) & = x^y \\\notag
    H\!R_4(x, y) & = e^{\log(x)^{e^y}} \\\notag
    H\!R_5(x, y) & = e^{e^{\log(\log(x))^{\exp^3(y)}}}

One of the first things to notice is that all my operators are expressible in closed form, unlike the the other hyper-operators. This means no hassle in their calculation, and no worries about how to write them. It also means no "new" operations. The second thing to notice is that the base b does not "get involved" in addition, multiplication, and exponentiation, so any base will interpolate these three operators. Another thing of note about them is that their extension to non-integer n is equivalent to the corresponding extension of tetration! This means if we can define tetration in a logical, consistent manner, then all these operators automatically get interpolated. Once we can calculate continuously iterated exponentials, then we can express all my operators as:

    H\!R_n(x, y) = \exp_b^{n-2}\left(\exp_b^{2-n}(x) \exp_b^{\binom{n-2}{2}}(y)\right)

where the continuously iterated exponential would allow n to be real or complex.