Let $\mathcal{E}_{\lambda}$ denote the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. Recall that $\mathcal{E}_{\lambda}$ is endowed with an algebraic operation $*$ defined by $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$.

Suppose that $X$ is a subalgebra of $\mathcal{E}_{\lambda}$, and suppose that $\gamma=\mathrm{crit}(j)$ for some $j\in X$. Furthermore, suppose that if $\simeq$ is a non-equality congruence on $X/\equiv^{\gamma}$, then there is some $k\in X$ with $\mathrm{crit}(k)<\mathrm{crit}(j)$ such that if $\delta=\mathrm{crit}(k)$ and $[r]_{\gamma}\simeq[s]_{\gamma}$, then $r\equiv^{\delta}s$. Then is the algebra $X$ necessarily isomorphic to some subalgebra of some classical Laver table $A_{n}$? Is the algebra $X$ embeddable into some quotient of some algebra of a classical Laver table $A_{n}$?


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