---
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 $x circle.small y = x y in Sigma^*$