Ok, but I'm sure more insight into a concrete model always improves the abstract one. Until now I only played with finite models (small finite sets of permutation/functions) or with monoid of functions over N.
I'm not on it seriously yet, I'm busy with some work. But I have few quick questions.
1- With closure you mean: let \( M \) be a monoid and \( B\subseteq M \) be a subset (a simple subset not a submonoid). With \( \overline{B} \) you mean the smaller submonoid of \( M \) that contains \( B \) right? In other words it is the submonoid generated by the set \( B \). In this case \( M={\mathcal C}^1({\mathbb R}) \).
2- Is your second condition \( \lim_{x\to+ \infty}f'(x)\in (1,+\infty] \)? Intuitively can I see this like the graph of f being "convex" in a neighborhood
of +infinity? Except for the +infinite case this seems a kind o "linearity at infinity".
3- the third condition seems to impose that the dynamics of our maps is very simple: it means that "all orbits come from the same source" and the source is external to our domain. Why do we need this? Is this to avoid fixed points and to avoid that superfunction gets a lower bound, thus non surjectivity?
Excuse me, always trivial things. I know.
Addendum 1: I guess the right contruction should be \( {\mathbb B}= \overline{B\cup B^{-1}} \). Given \( \psi\in {\mathbb B} \) then, by definition, \( \psi={\Omega_{i=0}^n}f_i \) for a finite n and \( f_i\in B\cup B^{-1} \). We have by anticommutativity of inversion \( \psi^{-1}={\Omega_{i=0}^n}f_{n-i}^{-1} \). This expression has meaning because by definition each \( f_i \) is invertible and \( f_i^{-1}\in B\cup B^{-1} \). We conclude that \( \psi^{-1}\in {\mathbb B} \) QED
Addendum 2: if \( B \) is a subset of \( {\mathcal C}^1({\mathbb R}) \) then \( B^{-1} \) is a subset of \( {\mathcal C}^1({\mathbb R}) \) iff elements of \( B \) are diffeomorphisms. I think the proof is trivial (but I should write it down). In that case \( {\mathbb B}= \overline{B\cup B^{-1}} \) is a subgroup of the group \( {\rm Diff}({\mathbb R}) \).
ps: Your pdf seems very excititng. I need to read it carefully.
I'm not on it seriously yet, I'm busy with some work. But I have few quick questions.
1- With closure you mean: let \( M \) be a monoid and \( B\subseteq M \) be a subset (a simple subset not a submonoid). With \( \overline{B} \) you mean the smaller submonoid of \( M \) that contains \( B \) right? In other words it is the submonoid generated by the set \( B \). In this case \( M={\mathcal C}^1({\mathbb R}) \).
2- Is your second condition \( \lim_{x\to+ \infty}f'(x)\in (1,+\infty] \)? Intuitively can I see this like the graph of f being "convex" in a neighborhood
of +infinity? Except for the +infinite case this seems a kind o "linearity at infinity".
3- the third condition seems to impose that the dynamics of our maps is very simple: it means that "all orbits come from the same source" and the source is external to our domain. Why do we need this? Is this to avoid fixed points and to avoid that superfunction gets a lower bound, thus non surjectivity?
Excuse me, always trivial things. I know.
Addendum 1: I guess the right contruction should be \( {\mathbb B}= \overline{B\cup B^{-1}} \). Given \( \psi\in {\mathbb B} \) then, by definition, \( \psi={\Omega_{i=0}^n}f_i \) for a finite n and \( f_i\in B\cup B^{-1} \). We have by anticommutativity of inversion \( \psi^{-1}={\Omega_{i=0}^n}f_{n-i}^{-1} \). This expression has meaning because by definition each \( f_i \) is invertible and \( f_i^{-1}\in B\cup B^{-1} \). We conclude that \( \psi^{-1}\in {\mathbb B} \) QED
Addendum 2: if \( B \) is a subset of \( {\mathcal C}^1({\mathbb R}) \) then \( B^{-1} \) is a subset of \( {\mathcal C}^1({\mathbb R}) \) iff elements of \( B \) are diffeomorphisms. I think the proof is trivial (but I should write it down). In that case \( {\mathbb B}= \overline{B\cup B^{-1}} \) is a subgroup of the group \( {\rm Diff}({\mathbb R}) \).
ps: Your pdf seems very excititng. I need to read it carefully.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)