Conway Chained Arrow Notation - Properties

Properties

  1. A chain of length 3 corresponds to Knuth's up-arrow notation and hyper operators:
    begin{matrix}
p to q to r = text{hyper}(p,r+2,q) = p !!! & underbrace{ uparrow dots uparrow } & !!! q = puparrow^r q.\
& !!! r text{ arrows} !!!
end{matrix}
  2. a chain XY is of the form Xp; hence:
  3. a chain starting with a is a power of a
  4. a chain 1 → Y is equal to 1
  5. a chain X → 1 → Y is equal to X
  6. a chain 2 → 2 → Y is equal to 4
  7. a chain X → 2 → 2 is equal to X → (X) (chain X with its value concatenated to it)

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