# Definition:Generator of Algebraic Structure

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## 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 $A$ if and only if $A$ is the algebraic substructure generated by $G$.

### Definition 2

The subset $G$ is a **generator** of $A$ if and only if:

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

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

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