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 $\left \langle {G}\right \rangle$.

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
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 $G$.

In this case, $G$ is the generator (or set of generators) of $\left({B, \circ}\right)$, or that $G$ generates $\left({B, \circ}\right)$.

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

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

Generator of a Semigroup
Let $\varnothing \subset X \subseteq S$, where $\left({S, \circ}\right)$ is a semigroup.

Then there exists $\left({T, \circ}\right)$, the smallest subsemigroup of $\left({S, \circ}\right)$ which contains $X$.

In this case, $X$ is the generator (or set of generators) of $\left({T, \circ}\right)$, or that $X$ generates $\left({T, \circ}\right)$.

$\left({T, \circ}\right)$ is the subsemigroup generated by $X$.

This is written $T = \left \langle {X} \right \rangle$.

This subsemigroup is proven to exist by Generator of a Semigroup.

Generator of a Group
If $\left({G, \circ}\right)$ is a group, then $H = \left\langle {X}\right\rangle$ is the subgroup of $\left({G, \circ}\right)$ generated by $X$.

If $X$ is a singleton, i.e. $X = \left\{{x}\right\}$, then we can (and usually do) write $T = \left\langle {x}\right\rangle$ for $T = \left\langle {\left\{{x}\right\}}\right\rangle$.

This subgroup is proven to exist by Generator of a Group.

Generator of a Ring
Let $\left({R, +, \circ}\right)$ be a ring.

Let $S \subseteq R$.

The subring generated by $S$ is the smallest subring of $R$ containing $S$.

Generator of an Ideal
Let $\left({R, +, \circ}\right)$ be a ring.

Let $S \subseteq R$.

The ideal generated by $S$ is the smallest ideal of $R$ containing $S$.

Generator of a Division Subring
Let $\left({D, +, \circ}\right)$ be a division ring.

Let $S \subseteq D$.

The division subring generated by $S$ is the smallest division subring of $D$ containing $S$.

Generator of a Field
Let $\left({F, +, \circ}\right)$ be a field.

Let $S \subseteq F$.

The field generated by $S$ is the smallest subfield of $F$ containing $S$.

Generator of a Module
Let $G$ be an $R$-module.

Let $S \subseteq G$.

The submodule generated by $S$ is the smallest submodule $H$ of $G$ containing $S$.

In this context, we say that $S$ is a generating set for $H$, or that $S$ generates $H$.

If we have $R$ as a field, we instead say S is a spanning set or that $S$ spans $H$.

This definition also applies when $G$ is a vector space.

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