Intersection of Subsemigroups/General Result/Proof 2

Theorem

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

Let $\mathbb H$ be a set of subsemigroups of $\struct {S, \circ}$, where $\mathbb H \ne \O$.

Then the intersection $\bigcap \mathbb H$ of the members of $\mathbb H$ is the largest subsemigroup of $\struct {S, \circ}$ contained in each member of $\mathbb H$.

Proof

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 $14.12$