A linear ordering on the quotient algebras of elementary embeddings?

We say that a finite self-distributive algebra $$(A,*)$$ is linear if there is some $$1\in A$$ where $$a*1=1,1*a=a$$ for all $$a\in A$$ and where if $$\preceq$$ is the relation where $$x\preceq y$$ if and only if $$x*a_{1}*...a_{n}=y$$ and $$x*a_{1}*...*a_{m}\neq 1$$ for $$0\leq m, then $$\preceq$$ is a linear ordering on $$A$$.

Let $$\mathcal{E}_{\lambda}$$ be the set of all elementary embeddings from $$V_{\lambda}$$ to $$V_{\lambda}$$. As always, let $$*$$ denote the operation on $$\mathcal{E}_{\lambda}$$ defined by $$j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$$. Then $$(\mathcal{E}_{\lambda},*)$$ satisfies the self-distributivity property $$j*(k*l)=(j*k)*(j*l)$$ and for each limit ordinal $$\gamma<\lambda$$, the relation $$\equiv^{\gamma}$$ defined by $$j\equiv^{\gamma}k$$ if and only if $$j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$$ for $$x\in V_{\gamma}$$ is a congruence on $$(\mathcal{E}_{\lambda},*)$$. The algebras $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ are always locally finite.

Are there any examples of closed subalgebras models of set theory where there is some cardinal $$\lambda$$ and subalgebras $$X\subseteq\mathcal{E}_{\lambda}$$ such that $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ is linear for all ordinals $$\gamma$$ but where $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ is not generated by a single element for sufficiently large limit ordinals $$\gamma$$?

• In the last sentence, I meant to say '$X/\equiv^{\gamma}$ is linear, but $X/\equiv^{\gamma}$ is not. . .' – Joseph Van Name Feb 21 at 22:27