### Tetration, Zeration For positive integers,
Multiplication is the same as repeated addition.
Exponentiation is the same as repeated multiplication.
There are two ways to go with this: what is repeated exponentiation? and what, repeated, is addition?
Repeated exponentiation is known as tetration.
The step below addition might be the increment operator (adding one.) Another possibility is called zeration. Both have some things going for them.

Any of these functions can be extended to work on real numbers. The picture is from the wikipedia article on tetration. I thought it was pretty. Mike Stay said…
a+b = b+a

If you pull back addition via exponentiation, you get multiplication.

a*b = b*a = exp(ln a + ln b)

If you pull that back, you get the Diffie-Hellman key exchange:

a#b = b#a = exp(ln a * ln b)

The process can be repeated indefinitely:

a\$b = b\$a = exp(ln a # ln b)
...

But you can also define an operator @ analogous to zeration by pushing forward addition:

a+b = b+a = exp(ln a @ ln b)
or
a @ b = ln(exp a + exp b)

It's related to the Zech logarithm:

zl(x) = ln(x+1)

since

ln(exp(a)+exp(b))
= ln((exp(a-b)+1) * exp(b))
= ln(exp(a-b)+1) + b
= zl(exp(a-b)) + b
= zl(exp(b-a)) + a by symmetry Mike Stay said…
http://tetration.itgo.com/paper.html Anonymous said…
a*b = b*a = exp(ln a + ln b)

You have written a homomorphism on the basis of the log of addition. It not Zeration which is defined under the classical scheme:

a o a o a o ... o a = a+b
repeat b Mike Stay said…
Yeah, I know. The one I described is the way to keep commutativity and associativity at every level, and clearly is not tetration. Matt said…
I like "succession" for zeration since I've seen the operator written as "succ"

Though zeration is a nice companion to tetration. Rodney said…
This is an awesome and fascinating jewel of mathatical modeling! Thanks bro, so random I found this today.. :-)
-Rodney