Definition:Generator of Algebraic Structure

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Definition

Let $\left({A, \circ}\right)$ 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:

  • $\forall x, y \in G: x \circ y \in A$;
  • $\forall z \in A: \exists x, y \in W \left({G}\right): z = x \circ y$

where $W \left({G}\right)$ 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 $A = \left \langle {G}\right \rangle$.


Also see

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