Intersection of Subsemigroups/General Result

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

Then the intersection $\bigcap \mathbb S$ of the members of $\mathbb S$ is itself a subsemigroup of $\struct {S, \circ}$.

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

Proof
Let $T = \bigcap \mathbb S$.

Then:

So by Subsemigroup Closure Test, $\struct {T, \circ}$ is a subsemigroup of $\struct {S, \circ}$.

Now to show that $\struct {T, \circ}$ is the largest such subsemigroup.

Let $x, y \in T$.

Then $\forall K \subseteq T: x \circ y \in K \implies x \circ y \in T$.

Thus $\forall K \in \mathbb S: K \subseteq T$.

Thus $T$ is the largest subsemigroup of $S$ contained in each member of $\mathbb S$.