Definition:Generator of Subsemigroup

Definition
Let $\varnothing \subset X \subseteq S$, where $\left({S, \circ}\right)$ is a semigroup.

Then there exists $\left({T, \circ}\right)$, the smallest subsemigroup of $\left({S, \circ}\right)$ which contains $X$.

In this case, $X$ is the generator (or set of generators) of $\left({T, \circ}\right)$, or that $X$ generates $\left({T, \circ}\right)$.

$\left({T, \circ}\right)$ is the subsemigroup generated by $X$.

This is written $T = \left \langle {X} \right \rangle$.

This subsemigroup is proven to exist by Generator of a Semigroup.