Definition:Generator of Subsemigroup

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


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 $X \subseteq T$.


$X$ 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.


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