Existence of Unique Subsemigroup Generated by Subset/Proof 2

Theorem

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

Let $\O \subset X \subseteq S$.

Let $\struct {T, \circ}$ be the subsemigroup generated by $X$.

Then $T = \gen X$ exists and is unique.

Proof

Let $\mathbb S$ be the set of all subsemigroups of $S$.

From Set of Subsemigroups forms Complete Lattice:

$\struct {\mathbb S, \subseteq}$ is a complete lattice.

where for every set $\mathbb H$ of subsemigroups of $S$:

the infimum of $\mathbb H$ necessarily admitted by $\mathbb H$ is $\ds \bigcap \mathbb H$.

Hence the result, by definition of infimum.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.12$