Generalized Kneser superfunction trick (the iterated limit definition)

[MphLee]

PROGRESS UPDATE (May 25, 2021):
I'll keep this as main hub for updates regarding superfunctions trick and superfunction-closed spaces.
I'm making progress on the work on the Jabotinsky-Nixon theory link seems not completely contained in this so I'll post those updates in the Jabotinky-Nixon thread.

I found some interesting trival results about groups that are extremely relevant to invertible superfunction-theory. I save them here for the future.
Now about superfunction closed space I'd like to restrict attention on the special case where the superfunction space is a group, i.e. every function in the space is invertible. No further assumption on the space.

We say that a group is superfunction-closed if \( \forall g,f\in G \) that are different from the identity do exists an element \( x\in G \) such that \( xf=gx \).

Remember: in a group, if an element is conjugated with the identity it IS the identity. In fact \( 1_Gx=xg \) implies \( 1_G=x^{-1}1_Gx=g \)


We can conclude that a non-trivial group \( G \) is supefunction-closed iff it has only two conjugacy classes: \( [1]_\sim =\{1\} \) and \( [g]_\sim =G/\{1\} \)

\( G=[1]_\sim \cup [g]_\sim \)  (g is a nontrivial element)

G being non-trivial means that exists \( g\in G \) s.t. \( g\neq 1_G \), i.e. exists a non-trivial element g. In other words this means that G has not less than 2 element.

What I found is that if G is finite and is supefunction closed then it only contains cyclic elements of order two. i.e.  \( g^2=1_G \). That means that the theory of ranks inside finite groups, when restricted to finite groups could finally provide examples for periodic sequences of subfunctions * (e.g. solution \( x,y \) to the Nixon's system of equations \( xf=yx \\yf=xy \) that dates back to the diamond operation thread).

Lets call an element a torsion element if  \( t^n=1_G \) for some non-zero n. For exemple, every involution is a torsion. The function i(x)=-x is a torsion in the group of bijections \( {\mathbb R}\to{\mathbb R} \); the function j(x)=1/x is a torsion in the group of bijections \( {\mathbb R}^+\to{\mathbb R}^+ \). So torsion elements have only a finite number of different iterates. Torsion free elements have infinite integer iterates.

Also if \( G \) is taken infinite and torsion-free, torsion free means that it has not torsion elements except the identity, then we can extend  \( G \) to a new group  \( \mathfrak G \) that is superfunction-complete using the HNN extension, an advanced group theoretic technique (actually Exercise 11.78 in Rotman's An Introduction to the Theory of Groups) that reminds somehow the procedure of adding layers after layers of new superfuntions like adding ranks going and taking the limit of this process.

Here some bookmarks for future studies: Embedding Theorems for Groups

[url=https://math.stackexchange.com/questions/773437/infinite-groups-with-only-2-conjugacy-classes?rq=1][/url]

* That statement is not accurate. When G is finite and super-closed then IT HAS ONLY TWO ELEMENTS. So the only periodic chain of subfunctions has lenght 1, i.e. it is a fixed point.