Definition:Generator

Definition

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

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

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

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

Let $\left({G, \circ}\right)$ be a group

Let $S \subseteq G$.

The subgroup of $\left({G, \circ}\right)$ generated by $S$ is the smallest subgroup $H$ of $G$ containing $S$.

This is denoted $H = \left\langle {S}\right\rangle$.

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

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 $S \subseteq F$ be a subset and $K \leq F$ a subfield.

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

The subring of $F$ generated by $K \cup S$, written $K[S]$, is the smallest subring of $F$ containing $K \cup S$.

The subfield of $F$ generated by $K \cup S$, written $K(S)$, is the smallest subfield of $F$ containing $K \cup S$.

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:

• $S$ a generating system for $H$ (over $R$)
• $S$ a generating set for $H$ (over $R$)
• $S$ generates $H$
• $S$ is a set of generators for $H$ (over $R$)
• $S$ is a generator for $H$ (over $R$)

If $R$ is a field, then:

• $S$ is a spanning set for $H$ (over $R$)

or

• $S$ spans $H$.

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

Generator of an Algebra

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$.

Let $S \subseteq A_R$ be a subset of $A_R$.

The subalgebra generated by $S$ is the smallest subalgebra $B_R$ of $A_R$ which contains $S$.

Notation

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