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 $$\left\langle {X, Y} \right\rangle$$.