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A ternary self-distributive algebra is an algebra $(X,t)$ that satisfies the identity $$t(u,v,t(x,y,z))=t(t(u,v,x),t(u,v,y),t(u,v,z)).$$ Is the variety of ternary self-distributive algebras generated by a set of finite algebras? Under the existence of a rank-into-rank cardinal, I have proven that the variety of binary self-distributive algebras is generated by its finite algebras. For the ternary case, I do not have much intuition about the situation.

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