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 $$\left({T_2, \circ}\right)$$.

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

$$a, b \in T$$

$$\Longrightarrow a, b \in T_1 \land a, b \in T_2$$ Definition of intersection

$$\Longrightarrow a \circ b \in T_1 \land a \circ b \in T_2$$ Subsemigroups are closed

$$\Longrightarrow a \circ b \in T$$ Definition of intersection

Thus $$\left({T, \circ}\right)$$ is closed, and is therefore a semigroup from the inheritance of associativity.