Kleene Star

languages

Definition

Kleene Star

$$\begin{array}{rl}{L}^{\ast }& =\bigcup _{n\ge 0}{L}^n=\{\epsilon \}\cup L^{+}\\ L^{+}& =\bigcup _{n\ge 1}{L}^n\end{array}$$

Identities

• $(A\cup \{\epsilon \})=A^{\ast }$
• $A\cdot A^{\ast }=A^{+}$
• $A^{+}\cup \{\epsilon \}=A^{\ast }$

Idempotence

$$(A^{\ast }{)}^{\ast }=A^{\ast }$$

Implications on Empty Strings

see empty string

• $\forall L:\epsilon \in L^{\ast }$
• $\epsilon \in L⟺\epsilon \in L^{+}$