Zeration

[bo198214]

a ° b = a + 1 , if a > b
a ° b = b + 1 , if a < b
a ° b = a + 2 = b + 2 , if a = b

This operation has no inverse, you can not even cancel it:
aob = cob as long as b>a,c

Also the above rules surely dont follow completely from
aoa=a+2

To be consistent with the other hyperoperations we would surely demand that
aoa=a+2
ao(aoa)=a+3
ao(ao(aoa))=a+4

or generally ao(a+n)=a+n+1 in analogy to \( a^{^n a} = {^{n+1}}a \) and \( a\times a^n = a^{n+1} \) and \( a+a\times n = a\times (n+1) \)

But this leads to a contradiction for n=0:
ao(a+0)=a+1 as opposed to aoa=a+2