# existence of maximal semilattice decomposition

Let $S$ be a semigroup^{}. A *maximal semilattice decomposition* for $S$ is a surjective^{} homomorphism^{} $\varphi :S\to \mathrm{\Gamma}$ onto a semilattice $\mathrm{\Gamma}$ with the property that any other semilattice decomposition factors through $\varphi $. So if ${\varphi}^{\prime}:S\to {\mathrm{\Gamma}}^{\prime}$ is any other semilattice decomposition of $S$, then there is a homomorphism $\mathrm{\Gamma}\to {\mathrm{\Gamma}}^{\prime}$ such that the following diagram commutes:

$$\text{xymatrix}S\text{ar}{[r]}^{\varphi}\text{ar}{[dr]}_{{\varphi}^{\prime}}\mathrm{\&}\mathrm{\Gamma}\text{ar}\mathrm{@}-->[d]\mathrm{\&}{\mathrm{\Gamma}}^{\prime}$$ |

###### Proposition.

Every semigroup has a maximal semilattice decomposition.

###### Proof.

Recall that each semilattice decompostion determines a semilattice congruence. If $\{{\rho}_{i}\mid i\in I\}$ is the family of all semilattice congruences on $S$, then define $\rho ={\bigcap}_{i\in I}{\rho}_{i}$. (Here, we consider the congruences^{} as subsets of $S\times S$, and take their intersection^{} as sets.)

It is easy to see that $\rho $ is also a semilattice congruence, which is contained in all other semilattice congruences.

Therefore each of the homomorphisms $S\to S/{\rho}_{i}$ factors through $S\to S/\rho $. ∎

Title | existence of maximal semilattice decomposition |
---|---|

Canonical name | ExistenceOfMaximalSemilatticeDecomposition |

Date of creation | 2013-03-22 13:07:12 |

Last modified on | 2013-03-22 13:07:12 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 5 |

Author | mclase (549) |

Entry type | Result |

Classification | msc 20M10 |

Defines | minimal semilattice congruence |

Defines | maximal semilattice decomposition |