Operations making non-context-free languages regular [closed]

1. If $$A$$ is a non-context free language, is $$A^*$$ a regular language?
2. If $$A$$ is a non-context free language and $$B$$ is a regular language, is it possible that their concatenation is a regular language?
• What do you think? What have you tried, and where did you get stuck? – Yuval Filmus Apr 15 '19 at 18:07

Let $$A$$ be $$\{a^p\mid p\text{ is prime}\}$$. Let $$B$$ be $$\{a\}^*$$.
Both $$A^*$$ and $$A\circ B$$ are regular. $$A^*$$ is $$\epsilon+aaa^*$$ and $$A\circ B$$ is $$aaa^*$$.
• Note that, if $A$ is a unary language (i.e., uses only a single letter alphabet) as in this case, then in fact the star $A^*$ is always regular. See this question: Show that the Kleene star of any unary language is regular. It seems the same holds for $A \cdot \{a\}^*$, because that language consists of all strings longer than the shortest in $A$. – Hendrik Jan Apr 11 '19 at 8:52