# Arriving at the critical points in an algebra of elementary embeddings in a unique way

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.

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 $$0\leq s? What about when $$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.

• 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. – Joseph Van Name Feb 6 at 2:41