Some "Theorem" on the generalized superfunction

[Leo.W]

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Leo.W Wrote

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The laws of both have something in common, like the anticommunity in multiplication and composition.
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Sure, I guess because the laws you describe are just informal observations. I claim that, if you try to formalize them, you'll come necessarily to the general laws I'm talking about.

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Note: n need not to be non-zero in this case. But in the Invariant Law it is crucial. Notice in the end that also the cancel law we are subject to even subtler technical details if we want a formal proof (probably we could only aim at an inclusion.).
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Hi, MphLee.


It's not very clear if you use bijective to a multivalued function is available?
Well, in fact, you'd better consider \( g^m [f,g] \) and \( [f,g]f^n \) and \( [f,g] \) are really equal to each other, since there lies the inverse multivalued function, and m and n could just be generalized into every nonzero complex constant...So this is what I mentioned, it's some kind of a...modern contradiction, you can't apply any laws that are not available to multivalued function to these . Though these laws are informal, I think it's still long before we can gradually accept the concept ''multivalued'', and only then can we get a formal definition of these laws...If it's convinient, you can consider the ''generalized Bottcher equation'' where f(z)=z+sqrt(z) and g(z)=z^2, when you try to generate the series coefficients in zeta, and if you never consider the other branch cut of f, f(z)=z-sqrt(z), the solution is only available in a subset of C, approximately {z|Re(z)>Im(z)^2}, you never get close to the cases beyond that.
The law 5 contains 2 parts, since I didn't make a classification. The former part is \( g^m [f,g] \) and \( [f,g]f^n \) and \( [f,g] \) are equal to each other. The latter part is symmetry law's generalization, for every nonzero t, \( f*h=f\to f^t*h=f^t \).
The symmetry law again, is something with self-contradiction, if you never consider multivalued function, for example, the symmetry cos(-z)=cos(z) never reaches t=-1, since arccos(z)+arccos(-z)=π, for most z, the symmetry of even functions are not available for their inverse if you never consider multivalued function. Of course, this law is not proved in the field of singlevalued functions.
I recommend to consider the law of symmetry as an approach to generate a specific branch cut from one to another.

If a solution set whose Z(z) is not a constant function has no elements, it's called pseudo-empty set(only my informal definition, maybe other terminology)
And it's pretty sure that there lies some pseudo-empty solution set, the most common example, consider f(z)=z and g(z)=a*z+b, obviously that Z(z)=a*Z(z)+b has only a constant solution. The empty solution set lies between most insymmetric functions and symmetric functions. It's unclear to me about how to figure out which of these are empty sets or pseudo-empty sets. And this problem can be something more relevant to corollary theorem. Well the first thing to notice, I think, is one of these are the identity function, so the solution of zeta is Z(z)=f^∞(z), a constant function.(sometimes blows up to infinity)Also, if f or g is constant, a pseudo-empty set. Another case is also easy to check, let f and g be different nonzeroth iteration of the same function, and this solution set may contain only constant solution(maybe infinity) or maybe no solutions at all, otherwise it must contain some function having a pseudo-periodic iteration: \( f^{s+t}~=f^s \).

Regards, Leo