Definition:Generated Algebraic Substructure

Definition
Let $\left({A, \circ}\right)$ be an algebraic structure.

Let $G \subseteq A$ be any subset of $A$.

Then there exists $\left({B, \circ}\right)$, the smallest substructure of $\left({A, \circ}\right)$ which contains $G$.

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

It is written $B = \left \langle {G} \right \rangle$.