An $(X,*,1)$ that satisfies the identities $x*(y*z)=(x*y)*(x*z),1*x=x,x*1=1$ is said to be a reduced Laver-like algebra if whenever $x_{n}\in X$ for $n\in\omega$, there is some $N\in\omega$ where $x_{0}*\dots*x_{N}=1$ (parentheses are grouped on the left $a*b*c=(a*b)*c$).
Suppose that $\mathcal{X}=((X_{n})_{n\in\omega},(\phi_{n,m})_{n\geq m})$ is an inverse system of reduced Laver-like algebras. Then we say that $\mathcal{X}$ is vast if
each transition mapping $\phi_{n,m}$ is surjective and for each $\phi_{n,m}$ there is some $c\in X_{n}$ where $c*c=1$ and where $\phi_{n,m}(x)=\phi_{n,m}(y)\Leftrightarrow c*x=c*y$, and
for each $m\in\omega$ and $x,y\in X_{m}$, there exists $n\geq m$ along with $x',y'\in X_{n}$ where $x'*y'\neq 1$.
Suppose that $(X,*,1)$ is a finite reduced Laver-like algebra generated by $k$ elements. Then does there exist a vast inverse system $((X_{n})_{n\in\omega},(\phi_{n,m})_{n\geq m})$ of Laver-like algebras where each $X_{n}$ is generated by $k$ elements?
The motivation behind this question comes from the fact that if $\mathcal{E}_{\lambda}$ denotes the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ and $j\in\mathcal{E}_{\lambda}\setminus\{1_{V_{\lambda}}\}$ and $\alpha_{n}=j^{n}(\mathrm{crit}(j))$ for $n\in\omega$, then $(\mathcal{E}_{\lambda}/\equiv^{\alpha_{n}})_{n\in\omega}$ is a vast system of Laver-like algebras, and the notion of a vast system of Laver-like algebras is an algebraization of $\mathcal{E}_{\lambda}$.
