There is famous Kołmogorov-Arnold theorem for continuous functions composition ( such function of several variables can be composed of continuous functions of two variables ). Put it as KA(continuous)

Specializaton of such theorem into smooth functions is false: there is no similar composition obeying smoothnes ( that is smooth function of several variables cannot be composition of smooth function of 2 variables). Put is as KA(smooth).

Take a lok here: Kolmogorov superposition for smooth functions

There is an obvious way around: generalization and even grand generalization.

Generalization: Is Kolmogorov-Arnold theorem true for just functions ( non-continuous), that is KA(function)?

If above question is interesting we may ask for Grand Generalization: Let KA(f, X) means the following hypothesis: given object containing several variables of some class ( function, relation, rules of inference etc), and characteristics X ( continuous, transitive etc) valid for such type, is it possible to compose every object of type f as finite number of objects of the same type with 2 variables with the same characteristics X?

Is there something known for Grand Generalization K(f,X) for various f and X?