Zeration

[tommy1729]

Not that I have become a " believer " but apart from

a[0]b = max(a,b) + 1

reference : {max,+} algebra , linear algebra.

a[0]b = ln( exp(a) + exp(b) )

reference : Bennet Hyperoperations, homomorphic defined operations and Tommy's distributive property.

Another thing makes sense , and will probably lead to the above 2 ... but if not that would be intresting ( I doubt it ! ) :

Rethinking : what is required for an operator [q] such that

a[q]b = b[q]a

and this is somehow related to iterations ?

Probably this equation :

Let N be a " neutral element " as a function of q.
Call this N_q.

then

a[q]b = b[q]a

implies :

f and g are some function :

f^[a - N_q](N_q[q]b) = g^[b - N_q](N_q[q]a)

that makes sense !

example [q] = +

a+b = b+a

"add+1"^[a - 0](0+b) = "add+1"^[b - 0](0+a)

example [q] = *

a*b = b*a

"add+b"^[a-1](1*b) = "add+a"^[b-1](1*a)

However notice that an expression like

f^[a - oo](...) does not make sense as a nonconstant function.

Also combining " all nice properties " seems impossible for zeration.

For instance max(a,b)+1 is not an analytic function.
Or a^^b =/= b^^a ...

regards

tommy1729