uni_notes/10 Courses/02 - SoSe 2026/AT/29593935 - Kleene Star & Kleene Plus.md

created, course, topic, related, type, status, tags

created	course	topic	related	type	status	tags
2026-04-08 10:15	29593850 - Automationtheory	kleen	29593929 - Alphabets	lecture	🟢	university

📌 Summary

[!abstract] Definition and Examples of the Kleene Star, the Kleene Plus and the Lemma Group Structure

---

📝 Content

Kleene Star

Denoted by Sigma^*. The Kleene Star (or Kleene operator or Kleene Closure) gives an infinite amount of 29593852 - Strings made up of the characters of the 29593929 - Alphabets 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 Lemma group structure is induced by the 29593935 - Kleene Star & Kleene Plus#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^*