# Obtaining linear orderings on the classical Laver tables from large cardinals

If $k:V_{\lambda}\rightarrow V_{\lambda}$ is an elementary embedding and $R\subseteq V_{\lambda}$, then let $k^{+}(R)=\bigcup_{\alpha<\lambda}k(R\cap V_{\alpha})$.

Recall that $\mathcal{E}_{\lambda}$ denotes the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Let $\mathcal{E}_{\lambda}[R]$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ such that $j^{+}(R)=R$. Then $\mathcal{E}_{\lambda}[R]$ is a subalgebra of the algebra of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$.

Suppose that $j:V_{\lambda}\rightarrow V_{\lambda}$ is a non-trivial elementary embedding with critical point $\kappa$. Let $A$ be a linear ordering of $V_{\kappa}$. Then for all $n$, we have $j^{n+1}(A)\cap V_{j^{n}(\kappa)}=j^{n}(A)$, so the sequence $(j^{n}(A))_{n\in\omega}$ of linear orderings is coherent. Therefore, define $j^{\omega}(A)=\bigcup_{n\in\omega}j^{n}(A)$ and let $B=j^{\omega}(A)$. Then $B$ is a linear ordering of $V_{\lambda}$ and $j\in\mathcal{E}_{\lambda}[B]$.

For each limit ordinal $\gamma<\lambda$, let $j\upharpoonright_{\gamma}:V_{\gamma+1}\rightarrow V_{\gamma+1}$ be the mapping where $j\upharpoonright_{\gamma}(x)=x\cap V_{\gamma}$ whenever $x\subseteq V_{\gamma}$.

Now define a linear ordering $<^{B}$ on $\mathcal{E}_{\lambda}[B]$ by letting $j<^{B}k$ precisely when $j\neq k$ and if $\gamma$ is the least limit ordinal such that $j\not\equiv^{\gamma+\omega}k$, then $(j\upharpoonright_{\gamma+\omega},k\upharpoonright_{\gamma+\omega})\in B$. Then the linear ordering $<^{B}$ on $\mathcal{E}_{\lambda}[B]$ induces a linear ordering $<^{B}$ on every quotient $\mathcal{E}_{\lambda}[B]/\equiv^{\gamma}$ by letting $[j]_{\gamma}<^{B}[k]_{\gamma}$ precisely when $[j]_{\gamma}\neq[k]_{\gamma}$ and $j<^{B}k$. Then the linear ordering $\leq^{B}$ on $\mathcal{E}_{\lambda}[B]/\equiv^{\gamma}$ is compatible with the algebraic structure of $\mathcal{E}_{\lambda}[B]/\equiv^{\gamma}$ in the sense that if $\mathfrak{j},\mathfrak{k},\mathfrak{l}\in\mathcal{E}_{\lambda}[B]/\equiv^{\gamma}$ and $\mathfrak{k}\leq^{B}\frak{l}$, then $\mathfrak{j}*\mathfrak{k}\leq^{B}\frak{j}*\frak{l}$.

Recall that the classical Laver table $A_{n}$ is the unique algebra $(\{1,\ldots,2^{n}\},*_{n})$ subject to the conditions $x*_{n}1=x+1\mod 2^{n}$ and $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ for all $x,y,z\in\{1,\ldots,2^{n}\}$. The algebras $A_{n}$ each contain a canonical linear ordering $\leq^{L}_{n}$ such that $x<^{L}_{n}y$ precisely when $(x)_{2^{n-1}}<_{n-1}^{L}(y)_{2^{n-1}}$ or ($(x)_{2^{n-1}}=(y)_{2^{n-1}}$ and $x<y$).

Recall that every one element generated subalgebra of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is isomorphic to a unique classical Laver table.

Let $j\in\mathcal{E}_{\lambda}$ be a non-trivial elementary embedding, and suppose now that $A$ is a linear ordering of $V_{\mathrm{crit}(j)}$. Now let $<_{n}^{j,A}$ be the unique linear ordering on $A_{n}$ such that if $\phi:A_{n}\rightarrow\langle j\rangle/\equiv^{\gamma}$ is an isomorphism, then $x<_{n}^{j,A}y$ precisely when $\phi(x)<^{j^{\omega}(A)}\phi(y)$. We say that the ordering $<_{n}^{j,A}$ is linear ordering induced by rank-into-rank embeddings.

The linear orderings $\leq_{n}^{L}$ and $\geq_{n}^{L}$ are easy to induce on $A_{n}$. What techniques can be used to induce linear orderings from rank-into-rank embeddings on $A_{n}$ besides $\leq_{n}^{L}$ and $\geq_{n}^{L}$? Which other linear orderings on $A_{n}$ are induced by rank-into-rank embeddings?