# Can a finitely generated algebra of rank-into-rank embeddings grow at rate $O(n\cdot\log(n))$?

Let $$\mathcal{E}_{\lambda}$$ be the set of all elementary embeddings from $$V_{\lambda}$$ to $$V_{\lambda}$$. If $$j\in\mathcal{E}_{\lambda}$$ is a non-trivial elementary embedding, then define $$\mathrm{crit}(j)$$ to be the least ordinal $$\alpha$$ where $$j(\alpha)>\alpha$$. Recall that $$\mathcal{E}_{\lambda}$$ may be endowed with a self-distributive operation $$*$$ where $$j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$$. We shall therefore regard $$\mathcal{E}_{\lambda}$$ as an algebraic structure with the operation $$*$$. Now, for each limit ordinal $$\gamma<\lambda$$, let $$\equiv^{\gamma}$$ be the congruence on $$\mathcal{E}_{\lambda}$$ where we set $$j\equiv^{\gamma}k$$ if and only if $$j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$$ for each $$x\in V_{\gamma}$$.

$$\mathbf{Theorem:}$$ If $$X$$ is a finitely generated subalgebra of $$\mathcal{E}_{\lambda}$$ and $$\gamma<\lambda$$ is a limit ordinal, then $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ is finite.

$$\mathbf{Theorem:}$$ If $$X$$ is a finitely generated subalgebra of $$\mathcal{E}_{\lambda}$$, then $$\{\mathrm{crit}(j)|j\in X\}$$ has order type $$\omega$$.

Suppose that $$X\subseteq\mathcal{E}_{\lambda}$$ is finitely generated. Let $$\mathrm{crit}_{n}(X)$$ denote the $$n$$-th element $$\{\mathrm{crit}(j)|j\in X\}$$. Then how slowly can the function $$n\mapsto|X/\equiv^{\mathrm{crit}_{n}(X)}|$$ grow? Can the function $$n\mapsto|X/\equiv^{\mathrm{crit}_{n+1}(X)}|-|X/\equiv^{\mathrm{crit}_{n}(X)}|$$ grow at a rate of $$O(\log(n))$$? The following result gives a lower bound of this growth rate of $$X$$.

$$\mathbf{Theorem:}$$ Suppose that $$X$$ is a finitely generated subalgebra of $$\mathcal{E}_{\lambda}$$. Then there is a $$\delta>0$$ such that for all $$n>0$$, we have $$|X/\equiv^{\mathrm{crit}_{n}(X)}|>\delta\cdot n\cdot\log(n).$$ (I have been able to obtain conjecturally sharp lower bounds for $$|X/\equiv^{\mathrm{crit}_{n}(X)}|$$ but it will probably be very difficult to actually prove these more precise lower bounds are sharp without extraordinary theoretical developments).

1. Does there exist a cardinal $$\lambda$$ and a finitely generated subalgebra $$X\subseteq\mathcal{E}_{\lambda}$$ along with a $$\delta>0$$ where for all $$n>0$$, we have $$|X/\equiv^{\mathrm{crit}_{n}(X)}|<\delta\cdot n\cdot\log(n)$$?

2. Does there exist a $$\lambda$$ and a finitely generated subalgebra $$X\subseteq\mathcal{E}_{\lambda}$$ along with a $$\delta>0$$ where for all $$n>0$$, we have $$|X/\equiv^{\mathrm{crit}_{n+1}(X)}|-|X/\equiv^{\mathrm{crit}_{n}(X)}|<\delta\cdot\log(n)$$?

3. Suppose that $$r>1$$. Does there exist a $$\delta>0$$ along with a cardinal $$\lambda$$ and $$X\subseteq\mathcal{E}_{\lambda}$$ is generated by $$r$$ elementary embeddings where if $$n>1$$ then $$|X/\equiv^{\mathrm{crit}_{n}(X)}|<\delta\cdot n\cdot\log(n)$$?

4. Suppose that $$r>1$$. Does there exist a $$\delta>0$$ along with a cardinal $$\lambda$$ and $$X\subseteq\mathcal{E}_{\lambda}$$ generated by $$r$$ elementary embeddings where if $$n>1$$ then $$|X/\equiv^{\mathrm{crit}_{n+1}(X)}|-|X/\equiv^{\mathrm{crit}_{n}(X)}|<\delta\cdot\log(n)$$?

5. If questions 1-4 are too difficult, can one prove assuming large cardinals that there is some $$\lambda$$ and finitely generated $$X\subseteq\mathcal{E}_{\lambda}$$ where if $$f(n)=|X/\equiv^{\mathrm{crit}_{n}(X)}|$$, then $$f(n)=O(a^{n})$$ for all $$a>1$$ or where $$f(n)=O(n^{d})$$ for some integer $$d$$?

I would be especially interested if the function mapping $$n$$ to the multiplication table of $$\mathrm{crit}_{n}(X)$$ is computable in polynomial time.

This question could more or less have been formulated algebraically without any reference to large cardinals. I will therefore gladly accept answers to the algebraic formulations of these questions.

Note: For all $$n$$, there exists algebras of elementary embeddings $$X\subseteq\mathcal{E}_{\lambda}$$ such that $$|X/\equiv^{\gamma}|=n$$ and where $$|\{\mathrm{crit}(j)|j\in X\}\cap(\gamma+1)|=n$$. The condition that $$X$$ is finitely generated is therefore necessary.

Note: My computer experiments suggest that it should be possible for $$|\langle j_{1},\dots,j_{r}\rangle/\equiv^{\mathrm{crit}_{n}(j_{1},\dots,j_{r})}|$$ to grow at a rate of $$O(n\cdot\log(n))$$.

Note: In case you are wondering, there is also an upper bound of the cardinality $$|\langle j_{1},\dots,j_{r}\rangle/\equiv^{\mathrm{crit}_{n}(j_{1},\dots,j_{r})}|.$$ We have $$|\langle j_{1},\dots,j_{r}\rangle/\equiv^{\mathrm{crit}_{n}(j_{1},\dots,j_{r})}|\leq\frac{r^{2^{n}}-1}{r-1}$$. This bound is sharp when we reformulate the question in an algebraic context and it is probably sharp in a set theoretic context as well.