Proof
(related to Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication)
The proof will consist of four steps:
This follows from the corresponding proposition.
This follows from the corresponding proposition.
Step 3: Demonstrate that \( ( \mathbb N\setminus\{0\}, \cdot )\) forms a commutative monoid.
- Step 1 shows that "\( \cdot \)" is associative, so \( ( \mathbb N\setminus\{0\}, \cdot )\) forms a
semigroup.
- Following the definition of multiplication, \(n \cdot 1=1\cdot n=n\).
- Thus, the element \(1\in\mathbb N\setminus\{0\}\) is the identity element of this semigroup.
- Together with step 2, this shows that \( ( \mathbb N\setminus\{0\}, \cdot )\) forms a commutative monoid.
Step 4: Show that \( ( \mathbb N\setminus\{0\}, \cdot )\) is cancellative.
This follows from the corresponding proposition.
These four steps show altogether that the set \( ( \mathbb N\setminus\{0\}, \cdot )\),
i.e. the set of natural numbers without the element \(0\), together with the multiplication
"\( \cdot \)" as a binary operation, forms a cancellative commutative monoid.
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References
Bibliography
- Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013