Let $A_{n}=(\{1,\ldots,2^{n}\},*_{n})$ be the algebra defined by $x*_{n}1=x+1\mod 2^{n}$ and $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ for all $x,y,z\in\{1,\ldots,2^{n}\}$. Suppose that $X$ is a subalgebra of some $A_{n}$ and $\simeq$ is a congruence on $X$. Then does there necessarily exist some $m$ along with some embedding $i:(X/\simeq)\rightarrow A_{m}$?

If so, then for all $n$, let $t_{n}$ be the least natural number $m$ such that whenever $X$ is a subalgebra of $A_{n}$ and $\simeq$ is a congruence on $X$, then there is some embedding $i:X/\simeq\rightarrow A_{m}$. Then what are some bounds on $t_{n}$? Or what is the exact value of $t_{n}$?


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