Monoid is not Empty

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Theorem

A monoid cannot be empty.


Proof

Let $\left({S, \circ}\right)$ be a the monoid.

By definition:

Identity: $\exists e_S \in S: \forall a \in S: a \circ e_S = a = e_S \circ a$


So a monoid must at least have an identity.

Therefore $e_S \in S$ and so $S$ is not the empty set.

$\blacksquare$


Sources