Is it true that a language is regular?

Question

Is it true that for every regular language $L \subseteq \{0,1\}^*$, language $\big\{ w^{|w|} \big| w \in L \big\}$ is regular? I don't know how to prove that.Could you give me a hint? Thank you

Answer 1

No, your language $W_L$ may fail pumping lemma because all the strings in $W_L$ must have length a perfect square. In fact a language with this property is regular if and only if it's finite: let $N$ be such that for all $x$ such that $\lvert x\rvert\ge N$ there are $u,v,w$ such that $uvw=x$, $\lvert uv\rvert\le N$, $\lvert v\rvert>0$ and $uv^jw$ is in the language for all $j\ge0$. Then the language should contain strings of length $\lvert u\rvert+\lvert w\rvert+ j\lvert v\rvert$ for all $j\ge0$, which is incompatible with all length of strings being perfect squares. This is unavoidable unless $N$ is such that there are no strings longer than $N$ in the language.