Definition:Generator

Definition
A generator of an algebraic structure $\left({A, \circ}\right)$ is a subset $G$ of the underlying set $A$ such that:

where $W \left({G}\right)$ is the set of words of $G$.
 * $\forall x, y \in G: x \circ y \in A$;
 * $\forall z \in A: \exists x, y \in W \left({G}\right): z = x \circ y$

That is, every element in $A$ can be formed as the product of a finite number of elements of $G$.

If $G$ is such a set, then we can write $A = \left \langle {G}\right \rangle$.

Finitely Generated
If an algebraic structure $\left({A, \circ}\right)$ has a generator of finite order, then $A$ is said to be finitely generated.

Generator of a Subset
The concept of a generator is usually defined in the context of particular types of structure, as follows.

Notation
We can also write $\left\langle {X \cup Y} \right\rangle$ as $\left\langle {X, Y} \right\rangle$.