[to do] fully iterative definition of goodstein HOS

[MphLee]

The last attempt seems too complicated to give low hanging fruits
I think it is better to start from this table

\[\begin{array}[|ccc|cc|]
&&&&{\rm ranks\, equiv.}&\\
\hline
{\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying\, goodstein} &B=1& A=\mathbb N\\
{\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying}\, A{\rm -equivariant\,  goodstein} &B=1& A\\
\hline
\mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B=\mathbb N&A=\mathbb N\\
\mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\,  goodstein} &B=\mathbb N&A\\
\hline
B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B&A=\mathbb N\\
B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\,  goodstein} &B&A\\
\hline
\end{array}\]
and refine it to a big picture.

Here the general scheme is: we take a map inside a certain category, from an object of ranks to an object of the support, and ask it to satisfy a condition, be it goodstein condition or some equivariant goodstein, or variant of it. All of this must be fixed by appropriate forgetfull functors \(U\) sending objects to the appropriate categories. From now on I'll identify the categories \(A{\rm - Act}\) and the functor (co-presheaves) category \([A,{\rm Set}]\) whenever \(A\) is a monoid and I'll omit the \(B\)-functor in order to keep notation clear. I'll use the common practice of denoting the hom-sets of a category \(\mathcal C\) by \(\mathcal C(X,Y)\) instead of \({\rm Hom}_{\mathcal C}(X,Y)\) except in the case of sets.


Set theoretic goodstein maps
This time I'll break the scheme differently, leaving aside everything that is \(A\)-equivariant since it need a qualitative upgrade that is intermediate to going categorical.

\[\begin{array}[|cccc|]
f{\rm maps\, in}&{\bf category}&{from\,the\,\,{\bf ranks}\,\, object}&{to\,the\,\,{\bf support}\,\, object}&&{\rm satisfying}&\\
\hline
{\bf h}\in{\rm Hom}(UJ,UM)&{in\,\,}1{\rm -Act} \,(of\,sets) &  (J\overset{+}{\to}J)\in \mathbb N{\rm - Act}& M\in{\rm Mon}&&\forall j\in UJ.\, {\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}&{\bf goodstein\, equation\,\, (GE)}\\
{\bf h}\in{\rm Hom}(UJ,UM)&{in\,\,}1{\rm -Act} \,(of\,sets) & (J\overset{-}{\to}J)\in \mathbb N{\rm - Act}& M\in{\rm Mon}&&\forall j\in UJ.\, {\bf h}_{j}s={\bf h}_{j^-}{\bf h}_{j}&{\bf anti}\!-\!{\rm GE\,\, (GE^-)}\\
{\bf h}\in{\rm Hom}(UJ,UG)&{in\,\,}1{\rm -Act} \,(of\,sets) &  (J\overset{+}{\to}J)\in \mathbb N{\rm - Act}& G\in{\rm Grp}&&{\bf h }\in {\rm fix}((\Sigma^G_S)_* \circ (+)^* )&{\bf GE\,\, over\, grps}\\
{\bf h}\in{\rm Hom}(UJ,UG)&{in\,\,}1{\rm -Act} \,(of\,sets) &  (J\overset{-}{\to}J)\in \mathbb N{\rm - Act}& G\in{\rm Grp}&&\forall j\in UJ.\, \Sigma^G_s({\bf h}_{j}^{-1})={\bf h}_{j^-}&{\bf  GE^-\,\, over\, grps}\\
\hline
\end{array}
\]


Special cases. since we just need monoid and groups that are non-abelian we can study set theoretic goodstein map on nice groups/monoids to obtain particular subtheories where is easier to obtain explicit results/computations. I believe I have proven that the theory over finite groups is trivial, but the problem is open over finite monoids. The theory over the  monoid \(\mathbb N^\mathbb N\) is basically extends recursion theory and gives hyperoperations. Also we can investigate the theory over multiplicative groups or rings or \(R\) algebras, i.e. \(R\)-modules \(X\) equipped with bilinear unital/associative maps \((-,-)_X:X\times X\to X\). Or, the most interesting case, by multiplication arising as multiplication in general linear groups of \(n\)-order matrices over a field (eg. finite fields). Here four interesting special cases that seems promising.

\[\begin{array}[|cccc|]
f{\rm maps\, in}&{\bf category}&{from\,the\,\,{\bf ranks}\,\, object}&{to\,the\,\,{\bf support}\,\, object}&&{\rm satisfying}&\\
\hline
A\in{\rm Hom}(U\mathbb N,{\rm End}(\mathbb N))&{in\,\,}1{\rm -Act} \,(of\,sets) &  (\mathbb N\overset{S}{\to}\mathbb N)\in \mathbb N{\rm - Act}& \mathbb N\in{\rm Set}&&A(n+1,x+1)=A(n,A(n+1,x))&{\bf Ackermann\, equation\,\,}\\
{\bf f}\in{\rm Hom}(UJ,UX)&{in\,\,}1{\rm -Act} \,(of\,sets) & (J\overset{+}{\to}J)\in \mathbb N{\rm - Act}& X\in R{\rm -Alg}&& ({\bf f}_{j},s)_X=({\bf f}_{j},{\bf h}_{j^+})_X&\\
f\in{\rm Hom}(UJ,R^{\times})&{in\,\,}1{\rm -Act} \,(of\,sets) &  (J\overset{+}{\to}J)\in \mathbb N{\rm -Act}& R\in{\rm Ring}&&f(j^+)\cdot s=f(j)\cdot f(j^+)&{\bf GE\,\, over\, non-comm.\,rings}\\
M\in{\rm Hom}(UJ,U{\sf GL}_n(k))&{in\,\,}1{\rm -Act} \,(of\,sets) &  (J\overset{+}{\to}J)\in \mathbb N{\rm -Act}& k\in{\rm Field}&& M_{j^+}SM_{j^+}^{-1}=M_j&k{\bf -linear\, GE\,\,}\\
\hline
\end{array}
\]

The set theoretic approach seems very rich yet limited. It boils down to these 4 cases, they can be reduced by two if the ranks dynamics is invertible because at that point goodstein and anti-goodstein equations give same solutions.

Dynamical goodstein maps
The first generalization appears spontaneously if we restrict the previous theory to groups.

Before generalizing further. Notice that the support object needs to induce a monoid operation over the Hom-set so we are pretty limited in extending the category from where we pick the support object. We are also limited in the choice of the ranks object. It can't just be a set because the goodstein equation ask us for a procedure that gives "the next rank", so it has to be at least a discrete dynamical system, or something equipped with an action of something that can be restricted to a discrete action: an object of a category equipped with an appropriate forgetfull functor \(F:\mathcal C\to\mathbb N{\rm -Act}\).

Generalizing. as I've noticed before, groups makes a cool phenomenon to appers. The anti-goodstein one does reduce in such a way that we can make it hold by structural means, without asking for it. We then use instead of the forgetfull functor, the functor \((G,s\in G) \mapsto \Sigma^G_s\) from pointed groups to dynamical systems sending a group and an element to the sub-function by s (it is functorial in \((G,s)\)).

This way we can naturally extend the scheme to \({\mathbb N}\)-equivariant goodstein maps with \(B\)-equivariant ranks. We take the ranks to be equipped with a \(B\)-action \(J\), for \(B\) a monoid equipped with an unit of time \(\upsilon\in B\), and as support object we take a \(B\)-action over the group \(G\) that when restricted to the natural numbers gives back \(\Sigma^G_s\): it is an element of the preimage category \({\boldsymbol \Sigma}\in (\upsilon^*)^{-1}(\Sigma^G_s)\subseteq B{\rm -Act}\) also expressible as the pullback, i.e. the fiber of the restriction functor bundle \(\upsilon^*\) that we can denote more comfortably as \(B{\rm -Act}_{\Sigma^G_s}\).

\[\begin{array}[|cccc|]
f{\rm maps\, in}&{\bf category}&{from\,the\,\,{\bf ranks}\,\, object}&{to\,the\,\,{\bf support}\,\, object}&&{\rm satisfying}&\\
\hline
{\bf h}\in\mathbb N{\rm -Act}(J,\Sigma^G_s)&{in\,\,}\mathbb N{\rm -Act}  &  J\in B{\rm -Act}& G\in{\rm Grp}&&nothing&{\bf  GE^-\,\,}by\, default\\
{\bf h}\in B{\rm -Act}(J,{\boldsymbol \Sigma})&{in\,\,}\mathbb N{\rm -Act} &  J\in B{\rm -Act}& {\boldsymbol \Sigma}\in B{\rm -Act}_{\Sigma^G_s}&&i.e. \, \forall \beta \in B,\,j\in J.\, {\bf h}_{\beta j}={\boldsymbol \Sigma}^\beta ({\bf h}_j)&B{\bf -equiv.\, ranks\, GE^-\,\,}by\, default\\
\hline
\end{array}
\]

This route seems even more promising but is much more limited than the previous one... I believe it is a dead end or maybe something bringing us to a totally different theory: the theory of group conjugation and its meaning.
I believe we should go another way and turning everything categorical.

Set-theoretic \(A\)-equivariant-goodstein maps
Here is the point where things gets harder at first but I've seen an opening for going functorial. Let \(A\) be a monoid and \(u\in A\) the unit of time.
The concept is simple: goodstein equation impose pointwise the condition of being \(\mathbb N\)-equivariant. We use the abstract identification of points \(x\in M\) of the support monoid to \(\mathbb N\)-iterations over \(M\), i.e. \[M\simeq {\rm Mon}(\mathbb N,M)\] and we use the same philosophy for seeing \(A\)-equivariant goodstein maps as selecting many \(A\)-iterations over \(M\) that by the \(A\)-equivariant goodstein equation (\(A\)-EGE) have their \(A\) equivariance imposed pointwise.

\[\begin{array}[|cccc|]
f{\rm maps\, in}&{\bf category}&{from\,the\,\,{\bf ranks}\,\, object}&{to\,the\,\,{\bf support}\,\, object}&&{\rm satisfying}&\\
\hline
{\bf h}\in {\rm Hom}(UJ,{\rm Mon}(\mathbb N,M))&{in\,\,}1{\rm -Act}  &  J\in \mathbb N{\rm -Act}& M\in{\rm Mon}&&\forall j\in UJ.\, {\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}&{\bf  GE\,\,}\\
{\bf h}\in {\rm Hom}(UJ,{\rm Mon}(A,M))&{in\,\,}1{\rm -Act} &  J\in \mathbb N{\rm -Act}& A,M\in{\rm Mon}&&i.e. \, \forall j\in UJ,\,a\in A.\, {\bf h}_{j^+}(u)s(a)={\bf h}_{j}(a){\bf h}_{j^+}(u)&A{\bf -EquivGE\,\,}\\
\hline
\end{array}
\]

Road to category theoretic goodstein maps. Here it is where we can spot the opening. First, note that until now the most lazy of our parameters was the object of ranks but since we can send dynamical system to sets and sets to discrete categories, embedding the category of sets in the category of categories is all we need. Secondly, the category of monoids can be enriched in cat: this means that the set of monoid homorphims is itself a category, a functor category \([A,M]\): namely  \[{\rm Ob}([A,M])={\rm Mon}(A,M)\] The reason I think this happens is because monoids themselves can be seen as one-object categories, monoid morphims as functors and natural transformations between two functors \(f,g:A\to M\) are exactly \(A\)-equivariances.

I find this spectacular. In symbols: let \(M(x,y)=\{\phi\in M:\, \phi x=y\phi\}\) then we have a bijection \[ M(x,y)\simeq {\rm Nat}_{[\mathbb N,M]}(x^\bullet,y^\bullet) \] where \(x^\bullet:\mathbb N\to A\) is the monoid morphism defined as \(x^0=1_M\) and \(x^{n+1}:=xx^n\).

In the same way, let \(f,g:A\to B\) be functor between monoid seen as categories. natural transformations \(\phi:f\implies g\) are elements \(\phi\in M\) that satisfy \(\forall a.\, \phi f(a)=g(a)\phi\)... so if \(M^A(f,g)\) is the set of such \(\phi\)s then we have a bijection \[M^A(f,g)\simeq {\rm Nat}_{[A,M]}(f,g) \]

The question remains... if we use functor categories as support objects... how we define on them the monoid operation on objects? t seems to be possible only when \(M\) is abelian or for some special \(A\) like the integers or the naturals.

TO BE CONTINUED