existence of maximal semilattice decomposition

Let S be a semigroupPlanetmathPlanetmath. A maximal semilattice decomposition for S is a surjectivePlanetmathPlanetmath homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕ:SΓ onto a semilattice Γ with the property that any other semilattice decomposition factors through ϕ. So if ϕ:SΓ is any other semilattice decomposition of S, then there is a homomorphism ΓΓ such that the following diagram commutes:


Every semigroup has a maximal semilattice decomposition.


Recall that each semilattice decompostion determines a semilattice congruence. If {ρiiI} is the family of all semilattice congruences on S, then define ρ=iIρi. (Here, we consider the congruencesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath as subsets of S×S, and take their intersectionMathworldPlanetmathPlanetmath as sets.)

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

Therefore each of the homomorphisms SS/ρi factors through SS/ρ. ∎

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