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00 Inbox/29593935 - Kleene.md

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+---
+created: 2026-04-08 10:15
+course: "[[29593850 - Automationtheory]]"
+topic: kleen
+related: "[[29593929 - Alphabets]]"
+type: lecture
+status: 🟢
+tags:
+  - university
+---
+## 📌 Summary
+
+> [!abstract]
+> 
+
+---
+
+## 📝 Content
+
+### Kleene Star
+Denoted by $Sigma^*$. The Kleene Star (or _Kleene operator_ or _Kleene Closure_) gives an infinite amount of strings made up of the characters of the alphabet $Sigma ^ *$. 
+$Sigma^*$ is the set of all string that can be generated by arbitrary concatenation of its characters.
+> $Sigma^* := union.big_(n>=0) A_n$
+> where $A_n$ is the set of all string combinations of length $n$
+
+#### Remarks
+- The same character can be used multiple times.
+- The empty string $epsilon$ is also part f $Sigma^*$.
+
+> [!Example]
+> $Sigma^* {a, b} = {epsilon, a, b, "aa", "ab", "ba", "bb", "aaa", "aab", ...}$
+
+> [!FACT]
+> - The set $Sigma^*$ is infinite, since we defined $Sigma$ to be non-empty.
+> - It is _countable_ and has the same cardinality as the set $NN$ of natural numbers
+
+### Kleene Plus
+The _Kleene Plus_ of an alphabet $Sigma$ is given by $Sigma^+ = Sigma^* backslash {epsilon}$ 
+
+### Lemma group structure
+The structure _Lemma_ is induced by the Kleene star - it is a monoid, that is a semigroup with a neutral element.
+
+> [!PROOF]
+> - Associativity has been shown
+> - Existence  of a neutral element has been shown.
+> - Closure under $circle.small$: Let $x in Sigma^*$ and $y in Sigma^*$ be two string over the alphabet $Sigma$. Then $x circle.small y = x y in Sigma^*$