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Is the variety of ternary self-distributive algebras generated by its finite members?

**Is the variety of ternary self-distributive algebras generated by its finite members?**

To convince you that this is a good question, let me explain a little about the mathematics behind this question.

Richard Laver has constructed a sequence $(A_{n})_{n\in\omega}$ of finite self-distributive algebras (which I now call the classical Laver tables), and he has shown under the existence of very large cardinals that the free self-distributive algebra on one generator embeds into $\varprojlim_{n\in\omega}A_{n}$. In particular, the free self-distributive algebra on one generator is contained in the variety generated by the classical Laver tables.

So I have extended the notion of the classical Laver table to a wide class of structures including the multigenic Laver tables. Furthermore, I have shown under strong large cardinal hypotheses that the free self-distributive algebra on countably many generators embeds into an inverse limit of multigenic Laver tables (the multigenic Laver tables are like the classical Laver tables but with multiple generators). In particular, the multigenic Laver tables generate the variety of all self-distributive algebras.

Now, the notion of self-distributivity can be generalized to $n$-ary self-distributivity and these generalized notions of self-distributivity give one an abstract notion of what it means for an algebraic structure to have “inner endomorphisms.” The notion of a Laver table can also be generalized to the notion of a ternary and $n$-ary Laver table. The only problem is that while the classical and multigenic Laver tables are always locally finite, the ternary Laver tables are in general not locally finite. The variety of ternary self-distributive algebras is probably generated by the ternary Laver tables, but since the ternary Laver tables are not locally finite, it is hard for me to predict whether the variety of self-distributive algebras is generated by its finite members.

thenone can open a separate meta post about the question. I find that this is a more efficient process :) $\endgroup$