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:

*the**hyper-operators*-- also known as:*hyperops*-- since it is a single short word,*hyper-operations*-- since there is little distinction between an*operator*(symbol) and*operation*(function),- higher hyper-operators -- since they generally produce values greater than other hyper-operators,
- right hyper-operators -- since they are defined by right-associative iteration (
`x`(`x`(`x``x`))), - the Ackermann function -- since this was defined by Wilhelm Ackermann in 1928,
- the Grzegorczyk hierarchy -- since this was defined by Andrzej Grzegorczyk in 1953,

*Ackermann function*is more common than*Grzegorczyk hierarchy*in English, and is by far the oldest term for these hyper-operators. These hyper-operators are what you get when you evaluate a repeated right-associative binary operation starting with addition.

*lower hyper-operators*-- also known as:- left hyper-operators -- since they are defined in terms of left-associative iteration (((
`x``x`)`x`)`x`),

- left hyper-operators -- since they are defined in terms of left-associative iteration (((
*mixed hyper-operators*-- also known as:- Bromer hyper-operators -- since Nick Bromer wrote about them in 1987,
- Müller hyper-operators -- since Markus Müller wrote about them in 1993,

*form a binary tree*.

*balanced hyper-operators*-- also known as:- balanced hyper-operators -- since they are balanced, and perfer neither left nor right associative iteration,
- centered hyper-operators -- since they generally produce values in the center of other hyper-operator values,

`x`^{x}which is a difficult function to iterate. Although he only talked about the iteration of`x`^{x}, he essentially paved the way for hyper-operators based on his approach, hence the name.

*commutative hyper-operators*-- not well-known. I suggest if you want to know more about them, read his paper about them here. As far as I understand, he defines addition and multiplication*on binary trees*, and iterates these operations instead. This is the most unique method I have seen to define hyper-operators. If only there was a conversion between binary trees and rationals...

*exponential hyper-operators*-- not known before, because I invented them. I know it is bad style to name things after yourself in math, but this is my website, so I can do what I want! :) These hyper-operators are defined in such a way as to include: addition, multiplication, and exponentiation, but not tetration, pentation, and so on.

Frappier hyper-operators are defined by the relation:

This definition has the following consequences:

It can be shown that the zeroth Frappier hyper-operator cannot exist, because by definition. Since this is a contradiction, we cannot have a zeroth Frappier hyper-operator.

Robbins hyper-operators are defined by the relation:

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

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

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