# Examples of self-distributing nilpotent algebras

Assume that $$(X, *, 1)$$ is an algebra that satisfies identities $$x * (y * z) = (x * y) * (x * z), x * 1 = 1.1 * x = x$$. Define the right powers by leaving $$x ^ {[1]} = x$$ and
$$x ^ {[n+1]} = x * x ^ {[n]}$$. We say that $$(X, *, 1)$$ is a nilpotent straight nilpotent self-distributive algebra so for each $$x in X$$, there is a $$n> 1$$ or $$x ^ {[n]} = 1$$. What are some examples of reduced self-distributive algebras of right nilpotent $$(X, *, 1)$$?

Non-examples Racks, Dosers, and Spindles (the Fuseaux are self-distributing algebras $$(X, *)$$ that satisfies idempotence identity $$x * x = x$$) with a cardinality greater than $$1$$ are examples of self-distributive algebras that are never nilpotent reduced self-distributive algebras. The nilpotent reduced self-distributive algebras must therefore have a strong irreversible form.

Preliminary examples: Here are some examples to start and make this problem more fun.

1. assume $$lambda$$ is a cardinal. Yes $$mathcal {E} _ { lambda}$$ is the set of all elementary nesting $$j: V _ { lambda} rightarrow V _ { lambda}$$, $$*$$ 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 incoherence, $$mathcal {E} _ { lambda} / equiv ^ { gamma}$$ is a right nilpotent reduced self-distributive algebra. This example can be generalized because there are finite algebras that look like $$mathcal {E} _ { lambda} / equiv ^ { gamma}$$ but that can not come from rank-in-row embeddings algebras.

2. Yes $$rightarrow$$ is Heyting's operation in a Heyting algebra with a maximum of element $$1$$then $$(X, rightarrow, 1)$$ is a right nilpotent reduced self-distributive algebra.

3. Assume that $$f: X rightarrow X$$ is a function such that $$f (1) = 1$$ and for all $$x in X$$, there is a $$n$$ or $$f ^ {n} (x) = 1$$. To define $$*$$ leaving $$x * y = f (y)$$. then $$(X, *)$$ is a right nilpotent reduced self-distributive algebra.