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Proof

(related to Corollary: Algebraic Structure of Strings over an Alphabet)

• Let $\Sigma^*$ be a non-empty set of all strings under the alphabet $\Sigma$.
• Let a concatenation be defined as its binary operation $\cdot: \Sigma^* \times \Sigma^* \mapsto \Sigma^*$.
• The concatenation is associative, i.e. for any letters $x,y,z\in \Sigma^*$ we have that $x(yz)=(xy)z$.
• Moreover, the empty string $\epsilon\in \Sigma^*$ is neutral with respect to concatenation.
• Therefore, \((\Sigma^*,\cdot)\) fulfills all axioms of a monoid.
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References

Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001