# Definition:Generator of Subsemigroup

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*This page is about generators of subsemigroups. For other uses, see Generator.*

## Definition

Let $\struct {S, \circ}$ be a semigroup.

Let $\O \subset X \subseteq S$.

Let $\struct {T, \circ}$ be the smallest subsemigroup of $\struct {S, \circ}$ such that $X \subseteq T$.

Then:

- $X$ is a
**generator**of $\struct {T, \circ}$ - $X$
**generates**$\struct {T, \circ}$ - $\struct {T, \circ}$ is the
**subsemigroup of $\struct {S, \circ}$ generated by $X$**.

This is written:

- $T = \gen X$

## Also known as

Some sources refer to such an $X$ as a **set of generators** of $T$, but this terminology is misleading, as it can be interpreted to mean that each of the elements of $X$ is itself a generator of $T$ independently of the other elements.

## Examples

### Positive Odd Numbers

Let $\struct {\Z, +}$ be the additive group of integers.

Let $A$ be the set of positive odd integers.

The subsemigroup of $\struct {\Z, +}$ generated by $A$ is the semigroup of strictly positive integers under addition.

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings