Tetration Forum
[2014] composition of 3 functions. - Printable Version

+- Tetration Forum (https://math.eretrandre.org/tetrationforum)
+-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)
+--- Thread: [2014] composition of 3 functions. (/showthread.php?tid=915)



[2014] composition of 3 functions. - tommy1729 - 08/25/2014

I want to consider composition of 3 nonlinear analytic functions.

Lets require that composition is noncommutative.

Denote composition with & and the functions with A,B,C.

A & B & C = B & C & A = C & A & B = T1

A & C & B = C & B & A = B & A & C = T2

T1 =/= T2.

This reminds me of quaternions and related stuff.
However we are considering functions here instead of numbers.

It seems like a natural continuation of algebra and dynamics.
Well this or a weaker set of equations.

An intresting weaker set of equations is

A & B & C = B & C & A = C & A & B = T1
{A & C & B , C & B & A , B & A & C }= T2

T1 is not equal to any member of the set T2.

I was inspired by being bored with
f^[a+b] = f^[a](f^[b]) = f^[b](f^[a])
or chebyshev type polynomials.

The commutative property started to feel like a prison for my mind.
We want more functions and concepts than just commutative.


regards

tommy1729