If $L$ is a regular language over $\Sigma = \{a, b\}$, which one of the following languages is NOT regular ?

Let us analyze the closure properties for each option:

• Option A: $L \cdot L^R$: Regular languages are closed under reversal ($L^R$ is regular) and concatenation. Thus, $L \cdot L^R$ is always regular.

• Option B: $\{ww^R \mid w \in L\}$: Here, every word $w \in L$ must be concatenated with its specific reverse $w^R$. If $L = \Sigma^* = (a+b)^*$, then $\{ww^R \mid w \in L\} = \{ww^R \mid w \in \{a,b\}^*\}$, which is the classic non-regular language of even-length palindromes (requires an unbounded stack memory to verify). Hence, this language is not necessarily regular.

• Option C: $\text{Prefix}(L)$: Regular languages are closed under the prefix operation.

• Option D: $\text{Suffix}(L)$: Regular languages are closed under the suffix operation.

Therefore, B is NOT regular.