# Definition:Generator of Semigroup

## 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.