semilattice decomposition of a semigroup
Semilattice decompositions arise from homomorphims of semigroups onto semilattices. If is a surjective homomorphism, then it is easy to see that we get a semilattice decomposition by putting for each . Conversely, every semilattice decomposition defines a map from to the indexing set which is easily seen to be a homomorphism.
A third way to look at semilattice decompositions is to consider the congruence defined by the homomorphism . Because is a semilattice, for all , and so satisfies the constraint that for all . Also, so that for all . A congruence which satisfies these two conditions is called a semilattice congruence.
Conversely, a semilattice congruence on gives rise to a homomorphism from to a semilattice . The -classes are the components of the decomposition.
|Title||semilattice decomposition of a semigroup|
|Date of creation||2013-03-22 13:07:09|
|Last modified on||2013-03-22 13:07:09|
|Last modified by||mclase (549)|