existence of maximal semilattice decomposition
Let be a semigroup. A maximal semilattice decomposition for is a surjective homomorphism onto a semilattice with the property that any other semilattice decomposition factors through . So if is any other semilattice decomposition of , then there is a homomorphism such that the following diagram commutes:
Every semigroup has a maximal semilattice decomposition.
Recall that each semilattice decompostion determines a semilattice congruence. If is the family of all semilattice congruences on , then define . (Here, we consider the congruences as subsets of , and take their intersection as sets.)
It is easy to see that is also a semilattice congruence, which is contained in all other semilattice congruences.
Therefore each of the homomorphisms factors through . ∎
|Title||existence of maximal semilattice decomposition|
|Date of creation||2013-03-22 13:07:12|
|Last modified on||2013-03-22 13:07:12|
|Last modified by||mclase (549)|
|Defines||minimal semilattice congruence|
|Defines||maximal semilattice decomposition|