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Full Version: [2014] composition of 3 functions.
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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.