Additional note. the problem of extending \(B\)-equivariance to the space of ranks need to be analyzed as follows. We need to keep track of where our maps lives. There are six cases of increasing complexity
\[\begin{array}[|ccc|cc|]
&&&&{\rm ranks\, equiv.}&\\
\hline
{\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying\, goodstein} &B=1& A=\mathbb N\\
{\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B=1& A\\
\hline
\mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B=\mathbb N&A=\mathbb N\\
\mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B=\mathbb N&A\\
\hline
B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B&A=\mathbb N\\
B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B&A\\
\hline
\end{array}\]
Classical case over monoids. In the case of \(M\) a monoid \(\mathbb N\)-equivariant goodstein map is a map that lives in the category \(1{\rm -Act}={\rm Set}\)
\[{\bf h}\in {\rm Hom}_{\rm Set}(J,{\rm Hom}_{\rm Mon}(\mathbb N,M))\]
that satisfies additional condition \({\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}\), i.e. each \({\bf h}_j\) is a map in \(\mathbb N{\rm -Act}={\rm Set}^{B\mathbb N}\).
Equivariant case over monoids. In the case the case of \(A\)-equivariant goodstein maps is a map that still lives in \(1{\rm -Act}={\rm Set}\)
\[{\bf h}\in {\rm Hom}_{\rm Set}(J,{\rm Hom}_{\rm Mon}(A,M))\]
but this time they satisfies \(\forall a\in A.\, {\bf h}_{j^+}(u)s(a)={\bf h}_{j}(a){\bf h}_{j^+}(u)\), i.e. each \({\bf h}_j\) is a map in \(A{\rm -Act}={\rm Set}^{BA}\).
In the case of \(M=G\) a group an incredible simplification becomes available for \(\mathbb N\)-equivariant goodstein maps: since \({\rm Hom}_{\rm Mon}(\mathbb N,G)\simeq G\) and on this set we can define the subfunction map \(\Sigma_s:G\to G\), sending \(g\in G\mapsto{ }gsg^{-1} \) we can upgrade the object \({\rm Hom}_{\rm Mon}(\mathbb N,G)\) from the category \(1{\rm -Act}={\rm Set}\) making it into an object \(\Sigma^G_S:=(G,\Sigma_s)\) of \(\mathbb N{\rm -Act}\), the same place where the space of ranks \(J=(J,(-)^-)\) lives. It turns out that the goodstein can be sinthetized by asking for maps that live in the category \(\mathbb N{\rm -Act}\).
Classical case over groups. In the case of \(M=G\) a group, a \(\mathbb N\)-equivariant goodstein map is just map that lives in the category \(\mathbb N{\rm -Act}\) of discrete dynamical systems
\[{\bf h}\in {\rm Hom}_{\mathbb N{\rm -Act}}(J,\Sigma_s^G)\]
Nothing more should be asked! This already implies it satisfies \({\bf h}_{j}s={\bf h}_{j^-}{\bf h}_{j}\).
Equivariant case over groups. Here we have a big obstacle since \({\rm Hom}_{\rm Mon}(A,G)\) is not equivalent to \(G\) in general and on this space of group homorphisms is not closed under the map \(\Sigma_s\) unless \(G\) is abelian... but in this case everything becomes trivial: even if it was it is not clear this is what we need to enforce the equivariant goodstein condition. The question is: what is the endofunction over \({\rm Hom}_{\rm Mon}(A,G)\) that we should consider and why? This is a big problem.
Maybe we can face this from a synthetic point of view. We need an object \(\mathfrak S(A,G)\in \mathbb N{\rm -Act}\) of the category \(\mathbb N{\rm -Act}\) associated with the group \(G\) that behaves as if it were the space of \(A\)-iterations over \(G\) and closed under taking subfunctions... something that is not possible as stated. We then look for maps
\[{\bf h}\in {\rm Hom}_{\mathbb N{\rm -Act}}(J,\mathfrak S(A,G) ) \]
And such that, given enough extra structure, we can somehow reconstruct the condition \(\forall a\in A. \Sigma_{s(a)}({\bf h}_{j}(u))={\bf h}_{j^-}(a)\).
Two more cases are to be studied and that completes the study of goodstein maps: the case with equivariant ranks but classical maps, and the case with everything equivariant.
\(B\)-Equivariant ranks and classical Goodstein over group. The first is again pretty straightforward. Just take any \(B\)-iteration of the map \(\Sigma_s:G\to G\), i.e. a map \({\boldsymbol \Sigma}:B\to G^G\) s.t. \( \forall \alpha,\beta\in B.\, \, {\boldsymbol \Sigma}(\alpha+\beta,g)={\boldsymbol \Sigma}(\alpha,{\boldsymbol \Sigma}(\beta,g))\) s.t. for some \(\upsilon \in B\) we have \({\boldsymbol \Sigma}(\upsilon,g)=gsg^{-1}\). And as space of ranks take an object \(J\in B{\rm -Act} \)
We look for maps \[{\bf h}\in {\rm Hom}_{B{\rm -Act}}(J,{\boldsymbol \Sigma})\]
This is actually iterating conjugation... it is really the way to access authentic non-integer ranks as started by Trappmann in his 2007 2008 thread: non-natural operation ranks.
But is hard... and we really should be interested when \(M\) is not a group... thus making everything harder.
\(B\)-Equivariant ranks and \(A\)-equivariant Goodstein over group. Here the model becomes highly non-trivial... I still don't know how to treat this... Probably the way to go is to upgrade this discussion, that is really at set theoretic level to categories. If we turn every object into a category... and every map into a functor... maybe all the problem presented here will dissolve.
\[\begin{array}[|ccc|cc|]
&&&&{\rm ranks\, equiv.}&\\
\hline
{\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying\, goodstein} &B=1& A=\mathbb N\\
{\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B=1& A\\
\hline
\mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B=\mathbb N&A=\mathbb N\\
\mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B=\mathbb N&A\\
\hline
B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B&A=\mathbb N\\
B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B&A\\
\hline
\end{array}\]
Classical case over monoids. In the case of \(M\) a monoid \(\mathbb N\)-equivariant goodstein map is a map that lives in the category \(1{\rm -Act}={\rm Set}\)
\[{\bf h}\in {\rm Hom}_{\rm Set}(J,{\rm Hom}_{\rm Mon}(\mathbb N,M))\]
that satisfies additional condition \({\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}\), i.e. each \({\bf h}_j\) is a map in \(\mathbb N{\rm -Act}={\rm Set}^{B\mathbb N}\).
Equivariant case over monoids. In the case the case of \(A\)-equivariant goodstein maps is a map that still lives in \(1{\rm -Act}={\rm Set}\)
\[{\bf h}\in {\rm Hom}_{\rm Set}(J,{\rm Hom}_{\rm Mon}(A,M))\]
but this time they satisfies \(\forall a\in A.\, {\bf h}_{j^+}(u)s(a)={\bf h}_{j}(a){\bf h}_{j^+}(u)\), i.e. each \({\bf h}_j\) is a map in \(A{\rm -Act}={\rm Set}^{BA}\).
In the case of \(M=G\) a group an incredible simplification becomes available for \(\mathbb N\)-equivariant goodstein maps: since \({\rm Hom}_{\rm Mon}(\mathbb N,G)\simeq G\) and on this set we can define the subfunction map \(\Sigma_s:G\to G\), sending \(g\in G\mapsto{ }gsg^{-1} \) we can upgrade the object \({\rm Hom}_{\rm Mon}(\mathbb N,G)\) from the category \(1{\rm -Act}={\rm Set}\) making it into an object \(\Sigma^G_S:=(G,\Sigma_s)\) of \(\mathbb N{\rm -Act}\), the same place where the space of ranks \(J=(J,(-)^-)\) lives. It turns out that the goodstein can be sinthetized by asking for maps that live in the category \(\mathbb N{\rm -Act}\).
Classical case over groups. In the case of \(M=G\) a group, a \(\mathbb N\)-equivariant goodstein map is just map that lives in the category \(\mathbb N{\rm -Act}\) of discrete dynamical systems
\[{\bf h}\in {\rm Hom}_{\mathbb N{\rm -Act}}(J,\Sigma_s^G)\]
Nothing more should be asked! This already implies it satisfies \({\bf h}_{j}s={\bf h}_{j^-}{\bf h}_{j}\).
Equivariant case over groups. Here we have a big obstacle since \({\rm Hom}_{\rm Mon}(A,G)\) is not equivalent to \(G\) in general and on this space of group homorphisms is not closed under the map \(\Sigma_s\) unless \(G\) is abelian... but in this case everything becomes trivial: even if it was it is not clear this is what we need to enforce the equivariant goodstein condition. The question is: what is the endofunction over \({\rm Hom}_{\rm Mon}(A,G)\) that we should consider and why? This is a big problem.
Maybe we can face this from a synthetic point of view. We need an object \(\mathfrak S(A,G)\in \mathbb N{\rm -Act}\) of the category \(\mathbb N{\rm -Act}\) associated with the group \(G\) that behaves as if it were the space of \(A\)-iterations over \(G\) and closed under taking subfunctions... something that is not possible as stated. We then look for maps
\[{\bf h}\in {\rm Hom}_{\mathbb N{\rm -Act}}(J,\mathfrak S(A,G) ) \]
And such that, given enough extra structure, we can somehow reconstruct the condition \(\forall a\in A. \Sigma_{s(a)}({\bf h}_{j}(u))={\bf h}_{j^-}(a)\).
Two more cases are to be studied and that completes the study of goodstein maps: the case with equivariant ranks but classical maps, and the case with everything equivariant.
\(B\)-Equivariant ranks and classical Goodstein over group. The first is again pretty straightforward. Just take any \(B\)-iteration of the map \(\Sigma_s:G\to G\), i.e. a map \({\boldsymbol \Sigma}:B\to G^G\) s.t. \( \forall \alpha,\beta\in B.\, \, {\boldsymbol \Sigma}(\alpha+\beta,g)={\boldsymbol \Sigma}(\alpha,{\boldsymbol \Sigma}(\beta,g))\) s.t. for some \(\upsilon \in B\) we have \({\boldsymbol \Sigma}(\upsilon,g)=gsg^{-1}\). And as space of ranks take an object \(J\in B{\rm -Act} \)
We look for maps \[{\bf h}\in {\rm Hom}_{B{\rm -Act}}(J,{\boldsymbol \Sigma})\]
This is actually iterating conjugation... it is really the way to access authentic non-integer ranks as started by Trappmann in his 2007 2008 thread: non-natural operation ranks.
But is hard... and we really should be interested when \(M\) is not a group... thus making everything harder.
\(B\)-Equivariant ranks and \(A\)-equivariant Goodstein over group. Here the model becomes highly non-trivial... I still don't know how to treat this... Probably the way to go is to upgrade this discussion, that is really at set theoretic level to categories. If we turn every object into a category... and every map into a functor... maybe all the problem presented here will dissolve.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)