# Is pumping lemma for regular languages “closed” against Kleene star?

If I have an infinite language $L$ which fulfills the Pumping lemma for regular languages, does $L^*$ also fulfill the same conditions?

• 1. if the first word of the Concatenation> n so that we can do a pumping – user6885 Feb 16 '13 at 15:30
• 2.word of the Concatenation<n we will Selected her to be V of the lemma and write her meny time we want – user6885 Feb 16 '13 at 15:32

In order to show the pumping property of a string like $x_1x_2 \dots x_n$, where each $x_i\in L$ you can distinguish two cases. Either $x_1$ is "short", which means you can pump $x_1$ completely and stay within $L^*$, or $x_1$ is "long" and you can pump "inside" $x_1$.
• Can't you always pump $x_1$ completely, meaning that for every language $L\subseteq \Sigma^*$, $L^*$ satisfies the pumping lemma? – Ariel Feb 6 '19 at 19:44
• @Ariel No, consider $\{a^nb^n\mid n\ge 1\}^*$. – Hendrik Jan Feb 6 '19 at 19:46
• This language does satisfy the pumping lemma. Actually, any language $L$ for which it holds that $L=L^*$ satisfies the pumping lemma (take the entire word), thus $L^*$ does seem to always satisfy the pumping lemma. – Ariel Feb 6 '19 at 19:50
• Nope. The Pumping Lemma (for regular languages) as I know it, see wikipedia, states that it is possible to find a pumpable segment within the first $p$ symbols of the string, where $p$ is the pumping length. This is not possible for, say $a^pb^p$. If you want to know more, I suggest to start a new question here. – Hendrik Jan Feb 6 '19 at 23:30