Semigroup is Subsemigroup of Itself

Theorem

Let $\struct {S, \circ}$ be a semigroup.

Then $\struct {S, \circ}$ is a subsemigroup of itself.

Proof

For all sets $S$, $S \subseteq S$, that is, $S$ is a subset of itself.

Thus $\struct {S, \circ}$ is a semigroup which is a subset of $\struct {S, \circ}$, and therefore a subsemigroup of $\struct {S, \circ}$.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 32$ Identity element and inverses