# semilattice decomposition of a semigroup

A semigroup $S$ has a *semilattice decomposition* if we can write $S={\bigcup}_{\gamma \in \mathrm{\Gamma}}{S}_{\gamma}$ as a disjoint union^{} of subsemigroups, indexed by elements of a semilattice $\mathrm{\Gamma}$, with the additional condition that $x\in {S}_{\alpha}$ and $y\in {S}_{\beta}$ implies $xy\in {S}_{\alpha \beta}$.

Semilattice decompositions arise from homomorphims of semigroups onto semilattices. If $\varphi :S\to \mathrm{\Gamma}$ is a surjective^{} homomorphism^{}, then it is easy to see that we get a semilattice decomposition by putting ${S}_{\gamma}={\varphi}^{-1}(\gamma )$ for each $\gamma \in \mathrm{\Gamma}$. Conversely, every semilattice decomposition defines a map from $S$ to the indexing set $\mathrm{\Gamma}$ which is easily seen to be a homomorphism.

A third way to look at semilattice decompositions is to consider the congruence^{} $\rho $ defined by the homomorphism $\varphi :S\to \mathrm{\Gamma}$. Because $\mathrm{\Gamma}$ is a semilattice, $\varphi ({x}^{2})=\varphi (x)$ for all $x$, and so $\rho $ satisfies the constraint that $x\rho {x}^{2}$ for all $x\in S$. Also, $\varphi (xy)=\varphi (yx)$ so that $xy\rho yx$ for all $x,y\in S$.
A congruence $\rho $ which satisfies these two conditions is called a *semilattice congruence*.

Conversely, a semilattice congruence $\rho $ on $S$ gives rise to a homomorphism from $S$ to a semilattice $S/\rho $. The $\rho $-classes are the components^{} of the decomposition.

Title | semilattice decomposition of a semigroup |
---|---|

Canonical name | SemilatticeDecompositionOfASemigroup |

Date of creation | 2013-03-22 13:07:09 |

Last modified on | 2013-03-22 13:07:09 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20M10 |

Defines | semilattice congruence |