Existence of Unique Subsemigroup Generated by Subset/Proof 2
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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 $14.12$