# Examples of nilpotent self-distributive algebras

Suppose that $$(X,*,1)$$ is an algebra that satisfies the identities $$x*(y*z)=(x*y)*(x*z),x*1=1,1*x=x$$. Define the right powers by letting $$x^{[1]}=x$$ and $$x^{[n+1]}=x*x^{[n]}$$. We say that $$(X,*,1)$$ is a right nilpotent reduced self-distributive algebra if for each $$x\in X$$, there is an $$n>1$$ where $$x^{[n]}=1$$. What are some examples of right nilpotent reduced self-distributive algebras $$(X,*,1)$$?

Non-examples Racks, quandles, and spindles (spindles are self-distributive algebras $$(X,*)$$ that satisfy the idempotence identity $$x*x=x$$) with cardinality greater than $$1$$ are examples of self-distributive algebras that are never right nilpotent reduced self-distributive algebras. The right nilpotent reduced self-distributive algebras must therefore have a strong form of irreversibility.

Preliminary examples: Here are a few examples to get things started and to make this problem more fun.

1. Suppose $$\lambda$$ is a cardinal. If $$\mathcal{E}_{\lambda}$$ is the set of all elementary embeddings $$j:V_{\lambda}\rightarrow V_{\lambda}$$, $$*$$ is defined by $$j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$$ $$\gamma$$ is a limit ordinal with $$\gamma<\lambda$$ and $$\equiv^{\gamma}$$ is the congruence on $$\mathcal{E}_{\lambda}$$ defined by $$j\equiv^{\gamma}k$$ if and only if $$j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$$ for each $$x\in V_{\gamma}$$. Then, by the Kunen inconsistency, $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ is a right nilpotent reduced self-distributive algebra. This example can be generalized since there are finite algebras that resemble $$\mathcal{E}_{\lambda}/\equiv^{\gamma}$$ but which cannot arise from the algebras of rank-into-rank embeddings.

2. If $$\rightarrow$$ is the Heyting operation in a Heyting algebra with maximum element $$1$$, then $$(X,\rightarrow,1)$$ is a right nilpotent reduced self-distributive algebra.

3. Suppose that $$f:X\rightarrow X$$ is a function such that $$f(1)=1$$ and for all $$x\in X$$, there is an $$n$$ where $$f^{n}(x)=1$$. Define $$*$$ by letting $$x*y=f(y)$$. Then $$(X,*)$$ is a right nilpotent reduced self-distributive algebra.