# Semigroup is Subsemigroup of Itself

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## 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