Suppose that there exists a rank-into-rank cardinal. Then let $R$ be the relation on the set of terms in the language of self-distributive algebras (or LD-monoids) on one generator where we set $R(s,t)$ if $\mathrm{crit}(s(j))\leq \mathrm{crit}(t(j))$. Let $E$ be the relation where $E(r,s,t)$ is true precisely when $r(j)\equiv^{\mathrm{crit}(t(j))}s(j)$. Let $T$ be the ternary relation where $T(r,s,t)$ precisely when there are elementary embeddings $j_{1},...,j_{n}$ with $$r(j)*j_{1}*...*j_{n}\equiv^{t(j)}s(j)$$ but where $$\mathrm{crit}(r(j)*j_{1}*...*j_{m})<\mathrm{crit}(t(j))$$ for $0\leq m<n$.

Under the existence of a rank-into-rank cardinal, the relations $R,E,T$ are all computable. Furthermore, the relations $R,E,T$ can be defined and computed in a purely algebraic context in terms of Laver tables under the assumption that the inverse limit of classical Laver tables $\varprojlim_{n}A_{n}$ contains free subalgebra on one generator, but this assumption has no known proof that does not use strong large cardinal assumptions.

1. What is the computational complexity of the relations $R,E,T$?

2. Is there any way to define $R,E,T$ without any reference to large cardinals and without any direct reference to inverse limits of finite self-distributive algebras?