Characterization about Kleene Closure, when is a language $L=L^*$?


What your argument actually shows is
$$
L = L^* Rightarrow L=L^2 Rightarrow L = L^+.
$$

Indeed, suppose that $L = L^*$. On the one hand, $L^2 subseteq L^* = L$. On the other hand, since $epsilon in L$, we have $L = epsilon L subseteq L^2$.

Next, suppose that $L = L^2$. Inductively, $L = L^n$. In particular, if $w in L^+$ then $w in L^n$ for some $n$, and so $w in L$, showing that $L^+ subseteq L$; and $L subseteq L^+$ trivially holds.

If $L = L^+$ then we cannot conclude that $L = L^2$. For example, $a^+ = (a^+)^+$ but $a^+ neq a^+a^+$.

If $L = L^2$ then we cannot conclude that $L = L^*$. For example, $emptyset = emptyset^2$ but $emptyset neq emptyset^*$.

However, it is possible to prove the following:

  • If $L = L^+$ and $epsilon in L$ then $L = L^2$. This shows that if $epsilon in L$, then $L = L^+ Leftrightarrow L = L^2$.
  • If $L neq emptyset$ then $L = L^2 Rightarrow L = L^*$. This shows that if $L neq emptyset$, then $L = L^* Leftrightarrow L = L^2$.

Indeed, if $L = L^+$ and $epsilon in L$ then $L^2 subseteq L^+ = L$, and $L subseteq epsilon L subseteq L^2$.

Next, if $L neq emptyset$ and $L = L^2$ then let $w$ be a word of smallest length in $L$. Since $L = L^2$, we can write $w = xy$, where $x,y in L$. By assumption, $|x|=|y|=|w|$, and so $|w| = 2|w|$, implying that $w = epsilon$. Together with $L^+ subseteq L$, this shows that $L^* subseteq L$, and so $L = L^*$.


Another nice characterization is:
$$ L = L^* Leftrightarrow L = M^* text{ for some language } M. $$
One implication is trivial. In the other direction, if $L = M^*$ then $L^* = (M^*)^* = M^* = L$.

This characterization remains true if we impose restrictions on $M$, for example $M cap M^2 = emptyset$. Indeed, suppose that $L = L^*$, let $tilde L = L setminus {epsilon}$, and let $M = tilde L setminus tilde L^2$. Note that $M^2 subseteq tilde L^2$, and so $M cap M^2 = emptyset$. Also, clearly $M^* subseteq tilde L^* = L$. On the other hand, we can prove inductively that $L subseteq M^*$.

Indeed, let $w in L$. If $w = epsilon$ then clearly $w in M^*$. Otherwise, $w in tilde L$. If $w in M$ then clearly $w in M^*$. Otherwise, $w in tilde L^2$, that is, $w = xy$, where $|x|,|y| < |w|$. By induction, $x,y in M^*$, and so $w = xy in M^*$.