In this paper, Laver claims (on page 14 of the Arxiv version) that if there exists a rank-into-rank embedding, then in some upward Easton forcing extension, there are elementary embeddings $j,k:V_{\lambda}\rightarrow V_{\lambda}$ with $j\neq k$ but where $j(\alpha)=k(\alpha)$ for each ordinal $\alpha$. In this model, what do the subalgebras $\langle j,k\rangle/\equiv^{\gamma}$ of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ look like? I am looking for a characterization of these algebras up-to-isomorphism. The algebra $\langle j,k\rangle/\equiv^{\gamma}$ is always finite by this answer.
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