Kleene Star

Let L : 𝒫 ⁢ ( Σ ∗ ) : 𝐿 𝒫 superscript Σ L:\mathcal{P}(\Sigma^{*}) be a Formal Language over Σ Σ \Sigma . The Kleene star of L 𝐿 L , denoted L ∗ superscript 𝐿 L^{*} , is the Join of the following Omega Chain:

L0superscript𝐿0\displaystyle L^{0} ={ε∣}\displaystyle=\left\{\;\varepsilon{\mid\;}\right\}
Ln+1superscript𝐿𝑛1\displaystyle L^{n+1} ={u⁢v}absent𝑢𝑣\displaystyle=\left\{\;uv\right\}

This can be readily computed as

L∗=⋃n:ℕLnsuperscript𝐿subscript:𝑛ℕsuperscript𝐿𝑛\par L^{*}=\bigcup_{n:\mathbb{N}}L^{n}