Are these two languages regular? Let $\operatorname{value}(x)$ be the result when the symbols of $x$ are multiplied from left to right according to
 $\qquad \displaystyle\begin{array}{c|ccc} 
    \times &amp; a &amp; b &amp; c \\
    \hline 
    a &amp; a &amp; a &amp; c \\
    b &amp; c &amp; a &amp; b \\
    c &amp; b &amp; c &amp; a
   \end{array}$

 Is $L=\{xy \mid |x|=|y| \land \operatorname{value}(x) = \operatorname{value}(y)\}$ regular?

 Is $L=\{xy \mid \operatorname{value}(x)= \operatorname{value}(y)\}$ regular?
Hint for (1): consider a different type of value, $\mathrm{value}(s) = 1$ if $s$ ends with $z$, and $\mathrm{value}(s) = 0$ otherwise. Define the language $L$ as in your case, and consider $L \cap x^* z y^* z$.

Hint for (2): consider the language $L_\alpha = \{xy : \mathrm{value}(x) = \mathrm{value}(y) = \alpha \}$. What is the relation between $L_\alpha$ and $M_\alpha = \{ x : \mathrm{value}(x) = \alpha \}$? Is $M_\alpha$ regular?