Group is Cancellable Monoid

Theorem

Let $\struct {G, \circ}$ be a group.

Then $\struct {G, \circ}$ is a cancellable monoid.


Proof

By definition, a group is a fortiori a monoid.

From Group Operation is Cancellable, $\circ$ is a cancellable operation in $G$.

Hence the result by definition of cancellable monoid.

$\blacksquare$


Sources

  • 1999: J.C. Rosales and P.A. García-Sánchez: Finitely Generated Commutative Monoids ... (previous) ... (next): Chapter $1$: Basic Definitions and Results
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