existence of maximal semilattice decomposition

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

 $\xymatrix{S\ar[r]^{\phi}\ar[dr]_{\phi^{\prime}}&\Gamma\ar@{-->}[d]\\ &\Gamma^{\prime}}$
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 ExistenceOfMaximalSemilatticeDecomposition 2013-03-22 13:07:12 2013-03-22 13:07:12 mclase (549) mclase (549) 5 mclase (549) Result msc 20M10 minimal semilattice congruence maximal semilattice decomposition