Monoid is not Empty

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

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids