Intersection of Subsemigroups

Theorem
Let $$\left({S, \circ}\right)$$ be a semigroup.

Let $$\left({T_1, \circ}\right)$$ and $$\left({T_2, \circ}\right)$$ be subsemigroups of $$\left({S, \circ}\right)$$.

Then the intersection of $$\left({T_1, \circ}\right)$$ and $$\left({T_2, \circ}\right)$$ is itself a subsemigroup of that $$\left({S, \circ}\right)$$.

If $$\left({T, \circ}\right)$$ is that intersection of $$\left({T_1, \circ}\right)$$ and $$\left({T_2, \circ}\right)$$, it follows that $$\left({T, \circ}\right)$$ is also a subsemigroup of both $$\left({T_1, \circ}\right)$$ and $$\left({T_2, \circ}\right)$$.

General Result
Let $$\mathbb S$$ be a set of subsemigroups of $$\left({S, \circ}\right)$$, where $$\mathbb S \ne \varnothing$$.

The intersection $$\bigcap \mathbb S$$ of the members of $$\mathbb S$$ is itself a subsemigroup of $$\left({S, \circ}\right)$$.

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

Proof
Let $$T = T_1 \cap T_2$$ where $$T_1, T_2$$ are subsemigroups of $$\left({S, \circ}\right)$$. Then:

$$ $$ $$ $$

Thus $$\left({T, \circ}\right)$$ is closed, and is therefore a semigroup from the Subsemigroup Closure Test.

The other results follow from this and Intersection Subset.

Proof of General Result
Let $$T = \bigcap \mathbb S$$.

Let $$T_k$$ be any element of $$\mathbb S$$. Then:

$$ $$ $$ $$

So $$\left({T, \circ}\right)$$ is a subsemigroup of $$\left({S, \circ}\right)$$.


 * Now to show that $$\left({T, \circ}\right)$$ 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 any $$K \in \mathbb S: K \subseteq T$$ and thus $$T$$ is the largest subsemigroup of $$S$$ contained in each member of $$\mathbb S$$.