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Aperiation is the operator made by Aarex that go further than up-arrows.

Normal Definition

How we go further than up-arrow notation? Simple; just add ◎ between the numbers, but we need the definition. a◎b is equal to a↑↑↑ ... ↑↑↑a with b ↑s!

Combining Symbols

It is easy to combine symbol. ◎↑ nests over ◎ symbol, ◎↑↑ nests over ◎↑ symbol, etc.

We also have ◎◎, which is the limit of ◎↑↑...; ◎◎◎, which is the limit of ◎◎↑↑...; etc.

There we go! But we need rules and definition:

█ is the rest or full of the expression, must have ↑ or/and ◎ symbols.

  • if █ = empty, a █ b = a*b
  • if █ have ↑ in the end, a █↑ b = a █ a █↑ b-1
  • if █ have ↑ in the end and b is 1, a █↑ 1 = a
  • if █ have ◎ in the end, a █◎ b = a █↑↑↑...↑↑↑ a with b ↑s.

Levels

We would have 2nd level of ◎, called as ◎2. ◎2 works the same as ◎, but replace ↑ by ◎.

Then we also have: ◎3by ◎2, ◎4by ◎3, ....

█ is the rest or full of the expression, must have ↑ or/and ◎x symbols.

  • if █ = empty, a █ b = a*b
  • if █ have ↑ in the end, a █↑ b = a █ a █↑ b-1
  • if █ have ↑ in the end and b is 1, a █↑ 1 = a
  • if █ have ◎0 in the end, a █◎0 b = a █↑ b
  • if █ have ◎n in the end, a █◎n b = a █◎n-1n-1n-1...◎n-1n-1n-1 a with b ◎n-1s.

Nesting Levels

It is easy to have up-arrow and ◎ levels.

█ is the rest or full of the expression, must have ↑ or/and ◎ symbols.

  • if █ = empty, a █ b = a*b
  • if █ have ↑ in the end, a █↑ b = a █ a █↑ b-1
  • if █ have ↑ in the end and b is 1, a █↑ 1 = a
  • if █ have ◎0 in the end, a █◎0 b = a █↑ b
  • if █ have ◎█↑ in the end, a █◎█↑ b = a █CCC...CCC a with b Cs, where C is ◎.

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