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.