# subsemigroup,, submonoid,, and subgroup

Let $S$ be a semigroup^{}, and let $T$ be a subset of $S$.

$T$ 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^{}.

$T$ is a *subgroup ^{}* of $S$ if $T$ is a submonoid which is a group.

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.

Title | subsemigroup,, submonoid,, and subgroup |
---|---|

Canonical name | SubsemigroupSubmonoidAndSubgroup |

Date of creation | 2013-03-22 13:02:03 |

Last modified on | 2013-03-22 13:02:03 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 5 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20M99 |

Related topic | Semigroup |

Related topic | Subgroup |

Defines | subsemigroup |

Defines | submonoid |

Defines | subgroup |