08/25/2014, 12:08 AM
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
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