Pseudo commutativity

Question

Operator commutativity is the basis for things like homomorphisms and linearity, e.g., $f(x+y) = f(x) + f(y)$.

Is there any meaning or development on a more general nature of this property? E.g., $f(x+y) = f(g(x)) + g(f(y)) = h(x) + h(y)$ or $f(x+g(y)) = h(f(x) * g(f(y))$ or various other relationships?

Or can this be handled under the standard models such categories with just added complexity?

What I'm interesting in understanding is if it is possible to transform maps that are not "linear" in to linear maps. Koopman and other variations show that all non-linearity is linear through expansion (possibly infinite).

Clearly linearity/homomorphisms are extremely useful due to their simplicity but they are also quite limiting. It is likely that many complex problems are complex due to our forcing a "square peg in a round hole" by trying to represent them in terms of linearity.

There seems to be no general means to deal with non-linearity (in the explicit analogous sense to the linear tools developed by mathematics). It seems that much of the machinery of "modern" mathematics is precisely to translate complex non-linear problems in to even more complex linear problems but which the complexity is "well regulated". (E.g., think of linearizing a system of non-linear differential equations in to a larger ("more complex") linear system.)

Does anyone know if there has been any investigation in to this specific area of "operational pseudo-commutativity" or is this really just dealt with with more complex arrangements of maps (e.g., category theory)? If so, does it resonate with mathematicians, in general, to say that higher level mathematics is precisely to deal with breaking down non-linear structures? After all, linear structure is pretty much solved in mathematics and in some sense can be represented in a single commuting square. More complex structures then should be said to be non-linear. In the sense of Koopman then we use these higher mathematical tools to "linearize" non-linear problems. Category theory could be thought of, then, as a tool to linearize non-linear problems much like how using substitution for differential equation is a tool to linearize non-linear differential equations or how manifold theory is a tool to linearize non-linear manifolds. It's almost as if all mathematics "after linearity" is to develop tools to map non-linearity in to linearity… it's as if it seems necessary (as what else could it be?).

The point of asking such a question is mainly that it seems by trying to linearize everything forces us to deal with an "infinite spectrum of linear problems" in the general case (e.g., Fourier series of non-differentiable functions). It seems that it might be better to try to transform non-linear problems in to other non-linear but tractable problems rather than try and brute force everything in to linear sub-problems.

Is there any branch of mathematics that is working along these directions specifically?

Answer 1

Various generalizations of the Cauchy equation $f(x+y)=f(x)+f(y)$, including extensions of the Pexider equation $f(x+y)=g(x)+h(y)$ to the multiplicative groupoid of matrices, are considered in the book Functional Equations in Several Variables by Aczel and Dhombres.

Chapter 19 of this book deals with so-called composite functional equations, such as $f(x+f(y))=f(y+f(x))$ or $f(x+f(x)y)=f(x)f(y)$; see many further references there.

More recently, there also was a characterization of groups of involutions by means of composite functional equations $f(xf(y))=f(f(x))y^{-1}$ and $f(xf(y))=y^{-1}f(f(x))$.