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Homesubsemigroup,, submonoid,, and subgroup

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# subsemigroup,, submonoid,, and subgroup

$T$ is a *subsemigroup* of $S$ if $T$ is closed under the operation of $S$; that it if $xy\in T$ for all $x,y\in T$.

$T$ is a *submonoid* of $S$ if $T$ is a subsemigroup, and $T$ has an identity element.

Note that submonoids and subgroups do not have to have the same identity element as $S$ itself (indeed, $S$ may not have an identity element). The identity element may be any idempotent element of $S$.

Let $e\in S$ be an idempotent element. Then there is a maximal subsemigroup of $S$ for which $e$ is the identity:

$eSe=\{exe\mid x\in S\}.$ |

In addition, there is a maximal subgroup for which $e$ is the identity:

$\mathcal{U}(eSe)=\{x\in eSe\mid\exists y\in eSe\;\text{st}\;xy=yx=e\}.$ |

Subgroups with different identity elements are disjoint. To see this, suppose that $G$ and $H$ are subgroups of a semigroup $S$ with identity elements $e$ and $f$ respectively, and suppose $x\in G\cap H$. Then $x$ has an inverse $y\in G$, and an inverse $z\in H$. We have:

$e=xy=fxy=fe=zxe=zx=f.$ |

Thus intersecting subgroups have the same identity element.

## Mathematics Subject Classification

20M99*no label found*

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