Definition:Generator of Algebraic Structure

Definition
Let $\struct {A, \circ}$ be an algebraic structure.

Let $G \subset A$ be a subset.

Definition 1
The subset $G$ is a generator of $A$ $A$ is the algebraic substructure generated by $G$.

Definition 2
The subset $G$ is a generator of $A$ :


 * $\forall x, y \in G: x \circ y \in A$;
 * $\forall z \in A: \exists x, y \in \map W G: z = x \circ y$

where $\map W G$ is the set of words of $G$.

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 $A = \gen G$.

Also see

 * Definition:Generated Algebraic Substructure

The concept of a generator is usually defined in the context of particular types of structure:


 * Definition:Generator of Magma
 * Definition:Generator of Semigroup
 * Definition:Generator of Monoid
 * Definition:Generator of Group