# Are these conditions sufficient for a self-distributive algebra to occur in the algebras of elementary embeddings?

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 $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$?