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:

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 $S\to S/\rho$.∎