Hyperoperation - Coincidence of Hyperoperations

Coincidence of Hyperoperations

Hyperoperations and are said to coincide on when . For example, for all, i.e. all hyperoperations above addition, . Similarly, but in this case both addition and mutiplication must be excluded. A point at which all hyperoperations coincide (excluding the unary successor function which does not really belong as a binary operation) is (2, 2) i.e. for all we have that . There is a connection between the arity of these functions i.e. two and this point of coincidence: since the second argument of a hyperoperation is the length of the list on which to fold the previous operation, and this is 2, we get that the previous operation is folded over a list of length two, which amounts to applying it to the pair represented by that list. Also, since the first argument is itself 2, and this is duplicated in the recursion, we arrive again at the pair (2, 2) with each recursion. This happens until we get to 2 + 2 = 4.

To be more precise, we have that = = . Note that the unit of need not be supplied to fold when the list has length > 1. To demonstrate this recursion by means of an example we take, which is two by itself twice i.e. . This, in turn is two plus itself twice i.e. . At +, the recursion terminates and we are left with four.