In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations) that starts with the unary operation of successor, then continues with the binary operations of addition, multiplication and exponentiation, after which the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration, pentation) and can be written using n-2 arrows in Knuth's up-arrow notation (if the latter is properly extended to negative arrow-indices for the first three hyperoperations). Each hyperoperation may be understood recursively in terms of the previous one by:
- with b occurrences of a on the right hand side of the equation
It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function:
This recursion rule is common to many variants of hyperoperations (see below).
Other articles related to "hyperoperation, hyperoperations":
... are algebraic structures equipped with at least one multi-valued operation, called a hyperoperation ... A hyperoperation (*) on a non-empty set H is a mapping from H × H to power set P*(H) (the set of all non-empty sets of H), i.e ... is a semihypergroup if (*) is an associative hyperoperation, i.e ...
... Hyperoperations and are said to coincide on when ... all hyperoperations above addition ... 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 ...