07/17/2022, 11:51 PM
Before I forget it, let me post here a note for the future.
As defined formally by me elsewhere (document in preparation), formal in the sense of deprived of interpretation/representation as endofunction, a formal (pre-)Goodstein sequence inside a pointed non-commutative monoid \((M,s)\) with ranks belonging to an \(\mathbb N\)-iteration \((J,{(-)}^+)\) is a function \({\bf h}:J\to M\) satisfying the system of \(\mathbb N\)-equivariance condition (aka superfucntion equations)
\[{\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}\]
Since after the latest discussions on the forum I have started to internalize and understand fully the relationship between being a superfucntion, being a family of superfunctions and being an iteration I believe that the previous version of the Goodstein f.equation shows itself as just the \(\mathbb N\)-equivariant version of a more general \(A\)-equivariant Goodstein f.equation.
Definition (\(A\)-equivariant (pre-)Goodstein equation): Fix the monoid of time \(A\) and an unit of time \(u\in A\). Take an \(A\)-pointed non-comm. monoid \((M,s)\), it will be our support and monoid morphism \(s:A\to M\) will be called the seed. Let \((J,{(-)}^+)\) be a an \(\mathbb N\)-action, called the space of ranks. An \(A\)-equivariant (pre-)Goodstein map is a map \({\bf h}:J\to {\rm Hom}_{\rm Mon}(A,M)\), i.e. a sequence of \(A\)-iterations/monoid homomorphisms \({\bf h}_j:A\to M\), over \(M\) indexed by \(J\) that satisfies the \(A\)-equivariant Goodstein f.equation over the seed \(s\) wrt the unit of time \(u\):
\[\forall a\in A.\,{\bf h}_{j^+}(u)s(a)={\bf h}_{j}(a){\bf h}_{j^+}(u)\]
Example: look for the special case \(A=\mathbb R\) adn \(u=1\), then this means the \(\mathbb R\)-equivariant goodstein functional equation doesn't ask the next hyperoperation to be just a superfunction of the previous but also to respect \(\mathbb R\)-iterations of the previous. This means the new definition is more close to the naive expectation of what we would like Goodstein hyperoperations to be. This means we have a sequence of \(\mathbb R\)-iterations \(f_j^t\) and that \[f^{\circ 1}_{j^+}\circ s^{\circ a}=f^{\circ a}_{j}\circ f^{\circ 1}_{j^+}\]
Open problem. some can clearly see that this is not perfection. The rank variable still belongs to the world of \(\mathbb N\)-iterations/actions. The ultimate Goodstein functional equation should be \(B\)-equivariant also i the rank variable... but how? The only way I can think of is by iterating group conjugation. We need to ask \(M\) to be a group. In this way, maybe we can find to make \(A\)-equivariant (pre-)Goodstein map \({\bf h}:J\to {\rm Hom}_{\rm Mon}(A,M)\) into a \(B\)-equivariant map, for some monoid \(B\) acting on the space of ranks.... but how?
As defined formally by me elsewhere (document in preparation), formal in the sense of deprived of interpretation/representation as endofunction, a formal (pre-)Goodstein sequence inside a pointed non-commutative monoid \((M,s)\) with ranks belonging to an \(\mathbb N\)-iteration \((J,{(-)}^+)\) is a function \({\bf h}:J\to M\) satisfying the system of \(\mathbb N\)-equivariance condition (aka superfucntion equations)
\[{\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}\]
Since after the latest discussions on the forum I have started to internalize and understand fully the relationship between being a superfucntion, being a family of superfunctions and being an iteration I believe that the previous version of the Goodstein f.equation shows itself as just the \(\mathbb N\)-equivariant version of a more general \(A\)-equivariant Goodstein f.equation.
Definition (\(A\)-equivariant (pre-)Goodstein equation): Fix the monoid of time \(A\) and an unit of time \(u\in A\). Take an \(A\)-pointed non-comm. monoid \((M,s)\), it will be our support and monoid morphism \(s:A\to M\) will be called the seed. Let \((J,{(-)}^+)\) be a an \(\mathbb N\)-action, called the space of ranks. An \(A\)-equivariant (pre-)Goodstein map is a map \({\bf h}:J\to {\rm Hom}_{\rm Mon}(A,M)\), i.e. a sequence of \(A\)-iterations/monoid homomorphisms \({\bf h}_j:A\to M\), over \(M\) indexed by \(J\) that satisfies the \(A\)-equivariant Goodstein f.equation over the seed \(s\) wrt the unit of time \(u\):
\[\forall a\in A.\,{\bf h}_{j^+}(u)s(a)={\bf h}_{j}(a){\bf h}_{j^+}(u)\]
Example: look for the special case \(A=\mathbb R\) adn \(u=1\), then this means the \(\mathbb R\)-equivariant goodstein functional equation doesn't ask the next hyperoperation to be just a superfunction of the previous but also to respect \(\mathbb R\)-iterations of the previous. This means the new definition is more close to the naive expectation of what we would like Goodstein hyperoperations to be. This means we have a sequence of \(\mathbb R\)-iterations \(f_j^t\) and that \[f^{\circ 1}_{j^+}\circ s^{\circ a}=f^{\circ a}_{j}\circ f^{\circ 1}_{j^+}\]
Open problem. some can clearly see that this is not perfection. The rank variable still belongs to the world of \(\mathbb N\)-iterations/actions. The ultimate Goodstein functional equation should be \(B\)-equivariant also i the rank variable... but how? The only way I can think of is by iterating group conjugation. We need to ask \(M\) to be a group. In this way, maybe we can find to make \(A\)-equivariant (pre-)Goodstein map \({\bf h}:J\to {\rm Hom}_{\rm Mon}(A,M)\) into a \(B\)-equivariant map, for some monoid \(B\) acting on the space of ranks.... but how?
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)