2 as hint was not sufficient I put in details.
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If I understand your question, you should consider $(a,aa)^*$$(a,aa)^* = \{ ( a^n, a^{2n} ) \mid n\ge 0 \}$.

Perhaps the following notation helps $(a,aa)^* = \{ \left(\begin{array}{c}a^n\\ a^{2n} \end{array}\right)\mid n\ge 0\}$. Interpreted over a two level alphabet this should be interpreted as $\{ \left(\begin{array}{c}a\\ a\end{array}\right)^n \left(\begin{array}{c}\$\\ a\end{array}\right)^n \mid n\ge 0\}$.

Your turn to conclude.

Here I assume \$'s can only be added at the end of the string.

If I understand your question, you should consider $(a,aa)^*$.

If I understand your question, you should consider $(a,aa)^* = \{ ( a^n, a^{2n} ) \mid n\ge 0 \}$.

Perhaps the following notation helps $(a,aa)^* = \{ \left(\begin{array}{c}a^n\\ a^{2n} \end{array}\right)\mid n\ge 0\}$. Interpreted over a two level alphabet this should be interpreted as $\{ \left(\begin{array}{c}a\\ a\end{array}\right)^n \left(\begin{array}{c}\$\\ a\end{array}\right)^n \mid n\ge 0\}$.

Your turn to conclude.

Here I assume \$'s can only be added at the end of the string.

1
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If I understand your question, you should consider $(a,aa)^*$.