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Proof

(related to Proposition: Algebraic Structure Of Natural Numbers Together With Addition)

The proof will consist of four steps:

Step 1: Demonstrate that the addition of natural numbers "\( + \)" is associative.

This follows from the corresponding proposition.

Step 2: Demonstrate that the addition of natural numbers "\( + \)" is commutative.

This follows from the corresponding proposition.

Step 3: Demonstrate that \( ( \mathbb N, + )\) forms a commutative monoid.

• Step 1 shows that "\( + \)" is associative, so \( ( \mathbb N, + )\) forms a semigroup.
• Because adding natural numbers is associative and because the natural number \(0\) is neutral with respect to addition, we have \(n + 0=0 + n=n\), thus the element \(0\in\mathbb N\) is the identity element of the semigroup.
• Because \( ( \mathbb N, + )\) forms a semigroup with an identity element (which is \(0\)), it is a monoid.
• Together with step 2, this shows that \( ( \mathbb N, + )\) forms a communtative monoid.

Step 4: Show that \((\mathbb N, +)\) is cancellative.

This follows from the corresponding proposition. These four steps show altogether that the set \((\mathbb N, + )\), i.e. the set of natural numbers, together with the addition "\( + \)" as a binary operation, forms a cancellative commutative monoid.

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References

Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013