# Can finite binary self-distributive algebras fit into small $n$-ary self-distributive algebras?

A binary operation $*$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$. An $n+1$-ary operation $t$ is said to be self-distributive if it satisfies the identity $$t(x_{1},...,x_{n},t(y_{1},...,y_{n},y)) =t(t(x_{1},...,x_{n},y_{1}),...,t(x_{1},...,x_{n},y_{n}),t(x_{1},...,x_{n},y)).$$

If $t$ is an $n+1$-ary self-distributive operation on a set $X$, then define the hull $\Gamma(X,t)$ to be the algebra $(X^{n},*)$ where $*$ is the binary operation such that $$(x_{1},...,x_{n})*(y_{1},...,y_{n})=(t(x_{1},...,x_{n},y_{1}),...,t(x_{1},...,x_{n},y_{n})).$$ Then the algebra $\Gamma(X,t)$ is always self-distributive.

Does there exist an $N$ such that for all finite binary self-distributive algebras $(X,*)$, there exists some $n$ and an $n+1$-ary self-distributive algebra $(Y,t)$ where $|Y|\leq N$ and where $(X,*)$ embeds into $\Gamma(Y,t)$?

• There is absolutely no reason for someone to downvote this question. Recently, people have been going around and downvoting and casting delete votes against nearly everything that I post. This is getting really old. – Joseph Van Name Mar 31 '19 at 2:38