Hyperoperation - Generalization

Generalization

For different initial conditions or different recursion rules, very different operations can occur. Some mathematicians refer to all variants as examples of hyperoperations.

In the general sense, a hyperoperation hierarchy is a family of binary operations on, indexed by a set, such that there exists where

• (addition),
• (multiplication), and
• (exponentiation).

Also, if the last condition is relaxed (i.e. there is no exponentiation), then we may also include the commutative hyperoperations, described below. Although one could list each hyperoperation explicitly, this is generally not the case. Most variants only include the successor function (or addition) in their definition, and redefine multiplication (and beyond) based on a single recursion rule that applies to all ranks. Since this is part of the definition of the hierarchy, and not a property of the hierarchy itself, it is difficult to define formally.

There are many possibilities for hyperoperations that are different from Goodstein's version. By using different initial conditions for or, the iterations of these conditions may produce different hyperoperations above exponentiation, while still corresponding to addition and multiplication. The modern definition of hyperoperations includes for all, whereas the variants below include, and .

An open problem in hyperoperation research is whether the hyperoperation hierarchy can be generalized to, and whether forms a quasigroup (with restricted domains).