# Definition:Generator of Semigroup

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## Contents

## Definition

Let $\left({S, \circ}\right)$ be a semigroup.

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

Let $\left({T, \circ}\right)$ be the smallest subsemigroup of $\left({S, \circ}\right)$ such that $X \subseteq T$.

Then:

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

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

## 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.

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 14$: Theorem $14.7$