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.
Does there exist a reasonable way to define $R,E,T$ that does not rely on results proven using large cardinals?
What is the computational complexity of the functions $R,E,T$?
Does the algebra $\langle j\rangle/\equiv^{t(j)}$ have a normal form in the following sense? Does there exist a reasonably efficiently computable family of functions $P_{t}:X\rightarrow\omega$ where $t$ ranges over the terms in the language of self-distributive algebras (or LD-monoids) such that $P_{t}(r)=P_{t}(s)$ if and only if $r(j)\equiv^{t(j)}s(j)$?
If the algebra $\langle j\rangle/\equiv^{t(j)}$ has an efficiently computable normal form (or at least the elements in $\langle j\rangle/\equiv^{t(j)}$ have strong invariants), then can the algebra $\langle j\rangle/\equiv^{t(j)}$ be used as a platform for Dehornoy's authentication scheme or the Kalka-Teicher key exchange? What about the baby Ko-Lee key exchange?
The baby Ko-Lee key exchange is the following key exchange. Suppose that $(X,\circ)$ is a semigroup and $x\in X$. The Alice and Bob produce an agreed upon secret piece of information among themselves by communicating over a public channel using the following steps:
i. Alice selects $a\in X$ and sends $r=a\circ x$ to Bob.
ii. Bob selects $b\in X$ and sends $s=x\circ b$ to Alice.
Let $K=a\circ x\circ b$.
iii. Alice can compute $K$ since $K=a\circ s$ and Alice knows $a,s$.
iv. Bob can compute $K$ since $K=r\circ b$.
$K$ is a piece of common information between Alice and Bob. One could generalize the baby Ko-Lee key exchange to a key-exchange for LD-monoids or LD-systems.
