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Theorem: Construction of Groups from Commutative and Cancellative Semigroups

Let \((H,\circ)\) be a semigroup, which is commutative and cancellative. Then there exists a unique group \((G,\ast)\) with the following properties:

1. There is a subset \(S\subset G\), which is a semigroup isomorphic to \(H\), i.e. where \((S,\ast)\simeq (H,\circ)\).
2. There is a subset \(S\subset G\), which is a semigroup isomorphic to \(H\), i.e. where \((S,\ast)\simeq (H,\circ)\).

Table of Contents

Proofs: 1

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Proofs: 2

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References

Bibliography

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