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Let $\mathcal{E}_{\lambda}$ denote the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. Then $\mathcal{E}_{\lambda}$ can be endowed with a self-distributive operation $*$ defined by $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. We shall therefore regard $\mathcal{E}_{\lambda}$ as an algebraic structure with operation $*$. To save space, all implied parentheses should appear on the left. i.e. $j*k*l*m=((j*k)*l)*m$. Define $\mathrm{crit}(j)$ to be the least ordinal $\alpha$ where $j(\alpha)\neq\alpha$.

$\mathrm{Theorem:}$ Suppose that $j_{n}\in\mathcal{E}_{\lambda}\setminus\{1_{V_{\lambda}}\}$ for $n\in\omega$. Then $\sup_{n\in\omega}\mathrm{crit}(j_{0}*...*j_{n})=\lambda$.

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

If $j_{1},...,j_{r}\in\mathcal{E}_{\lambda}$, then let $\mathrm{crit}_{n}(j_{1},...,j_{r})$ denote the $r$-th critical point in the set $\{\mathrm{crit}(j)|j\in\langle j_{1},...,j_{r}\rangle\}$.

$\mathrm{Theorem:}$ If $j_{1},...,j_{r}\in\mathcal{E}_{\lambda}$, then for all $n$ there is a sequence $a_{1},...,a_{v}$ of elements in $\{1,...,r\}$ where $\mathrm{crit}(j_{a_{1}}*...*j_{a_{v}})=\mathrm{crit}_{n}(j_{1},...,j_{r})$ but where $\mathrm{crit}(j_{a_{1}}*...*j_{a_{u}})<\mathrm{crit}_{n}(j_{1},...,j_{r})$ whenever $0<u<v$.

Does there exist a cardinal $\lambda$ and an $r>1$ along with elementary embeddings $j_{1},...,j_{r}\in\mathcal{E}_{\lambda}$ such that for all $n$, there exists a unique sequence $a_{n,0},...,a_{n,s_{n}}$ of numbers in $\{1,...,r\}$ where $$\mathrm{crit}(j_{a_{n,0}}*...*j_{a_{n,s_{n}}})=\mathrm{crit}_{n}(j_{1},...,j_{r})$$ but where $$\mathrm{crit}(j_{a_{n,0}}*...*j_{a_{n,s}})<\mathrm{crit}_{n}(j_{1},...,j_{r})$$ whenever $0\leq s<s_{n}$? What about when $j_{1},...,j_{r}$ extend to elementary embeddings from $V_{\lambda+1}$ to $V_{\lambda+1}$.

This question could more or less be restated in a purely algebraic context in several different ways, so answers to the versions of this question in an algebraic context are acceptable too.

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  • $\begingroup$ So I have been doing recent computer experiments with Laver-like algebras and it seems like the finite algebras that have the "unique arrival to critical point property" with respect to two generators are the easiest two generator Laver-like algebras to create which are novel in the sense that they generate new multigenic Laver tables and new critically simple Laver-like algebras. There is definitely no shortage of Laver-like algebras on two generators with the unique arrival to critical point property. $\endgroup$ – Joseph Van Name Feb 6 at 2:41

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