Suppose that $\mathcal{E}_{\lambda}$ is the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ and $*$ is the operation on $\mathcal{E}_{\lambda}$ defined by $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. If $\gamma$ is a limit ordinal with $\gamma<\lambda$, define an equivalence relation $\equiv^{\gamma}$ on $\mathcal{E}_{\lambda}$ by letting $j\equiv^{\gamma}k$ if and only if $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$. Then $\equiv^{\gamma}$ is a congruence on $\mathcal{E}_{\lambda}$. Now, a natural question to ask is if there is a natural characterization of the subalgebras and quotient algebras of the algebras that may arise as $\mathcal{E}_{\lambda}/\equiv^{\gamma}.$

Let us now attempt an axiomatization of the subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ that does not require set theory.

An algebra $(X,*)$ is said to be self-distributive if $x*(y*z)=(x*y)*(x*z)$ for all $x,y,z\in X$. A self-distributive algebra $(X,*)$ is said to be a reduced Laver-like algebra if there is some $1\in X$ where $1*x=x,x*1=1$ for $x\in X$ and where if $x_{n}\in X$ for each $n\in\omega$, then there is some $N\in\omega$ with $x_{0}*\dots*x_{N}=1$ (here parentheses are grouped on the left, so $a*b*c=(a*b)*c$).

$\mathbf{Theorem:}$ If $\lambda$ is a cardinal and $\gamma$ is a limit ordinal with $\gamma<\lambda$, then $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is Laver-like.

reduced Laver-like algebras behave a lot like the algebras $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ in the sense that Laver-like algebras have a notion of a critical point, a composition operation, and an analogue of $\equiv^{\gamma}$. Furthermore, the notion of a critical point for Laver-like algebras behaves in many ways the same as it does in $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ and the notion of a critical point is the magic that makes Laver-like algebras run.

If $(X,*)$ is a reduced Laver-like algebra, then there is a unique ordinal $\alpha$ and function $\mathrm{crit}:X\rightarrow\alpha+1$ such that

  1. $\mathrm{crit}(x)=\alpha$ if and only if $x=1$,

  2. $\mathrm{crit}(y)=\mathrm{crit}(x*y)$ whenever $\mathrm{crit}(y)<\mathrm{crit}(x)$, and

  3. $\mathrm{crit}(y)<\mathrm{crit}(x*y)$ whenever $\mathrm{crit}(x)\leq\mathrm{crit}(y)<\alpha$.

$\mathbf{Theorem:}$ Suppose that $j\in\mathcal{E}_{\lambda}$. Then $(j*j)(\alpha)\leq j(\alpha)$ whenever $\alpha<\lambda$.

$\mathbf{Corollary:}$ If $X$ is isomorphic to a quotient of a subalgebra of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$, then $\mathrm{crit}(x*x*y)\leq\mathrm{crit}(x*y)$ for all $x,y\in X$.

$\mathbf{Example:}$ There exists an $11$ element reduced Laver-like algebra $X$ along with $x,y\in X$ where $\mathrm{crit}(x*x*y)>\mathrm{crit}(x*y)$. Therefore, $X$ is not isomorphic to any quotient of a subalgebra of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$.

$\mathbf{Question:}$ If $(X,*)$ is a finite reduced Laver-like algebra such that $\mathrm{crit}(x*x*y)>\mathrm{crit}(x*y)$, then is $X$ isomorphic to some subalgebra of some $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ in some model of ZFC? If not, then what is the correct purely algebraic characterization of the finite subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ or at least the finite quotients of subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$?


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