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 $G$.

Let $\left({B, \circ}\right)$ be the smallest substructure of $\left({A, \circ}\right)$ such that $G \subseteq B$.

Then:
 * $G$ is a generator of $\left({B, \circ}\right)$
 * $G$ generates $\left({B, \circ}\right)$
 * $\left({B, \circ}\right)$ is the substructure of $\left({A, \circ}\right)$ generated by $G$.

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

Also known as
Some sources refer to such a $G$ as a set of generators of $B$, but this terminology is misleading, as it can be interpreted to mean that each of the elements of $G$ is itself a generator of $B$ independently of the other elements.

Also see

 * Definition:Generator