Zeration

[bo198214]

a o a=a+2

Though this is intuitive - because true for all operations [n] with n>0 on the left side - this is not true for zeration - if we require the following (mother) law to hold:

a [n+1] (x+1) = a [n] ( a [n+1] x )

which you already used on the left side for n=0, x=0; [0] is zeration, [1] is addition.

Actually not only *you* were confused about the failure of (*) a [n+1] m = a [n] ( a [n] ( ....( a[n]a ) ) ) - right side containing m times a - for n=0. But if one takes a close look one sees that one can only expect (from the mother law):
a [n+1] m = a [n] ( a [n] ( ... a [n] ( a [n+1] 1 ) ) ) ), where the right side contains m times a.

if now a [n+1] 1 = a (which is the case for all n>0) then (*) follows. But n=0 remains the exception, because a [0+1] 1 = a+1 != a.