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 = p\uparrow^r q.\\ & \!\!\! r \text{ arrows} \!\!\! \end{matrix}
  2. a chain X → Y is of the form X → p; 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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