semilattice decomposition of a semigroup

A semigroup S has a semilattice decomposition if we can write S=γΓSγ as a disjoint unionMathworldPlanetmathPlanetmath of subsemigroups, indexed by elements of a semilattice Γ, with the additional condition that xSα and ySβ implies xySαβ.

Semilattice decompositions arise from homomorphims of semigroups onto semilattices. If ϕ:SΓ is a surjectivePlanetmathPlanetmath homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, then it is easy to see that we get a semilattice decomposition by putting Sγ=ϕ-1(γ) for each γΓ. Conversely, every semilattice decomposition defines a map from S to the indexing set Γ which is easily seen to be a homomorphism.

A third way to look at semilattice decompositions is to consider the congruencePlanetmathPlanetmathPlanetmathPlanetmath ρ defined by the homomorphism ϕ:SΓ. Because Γ is a semilattice, ϕ(x2)=ϕ(x) for all x, and so ρ satisfies the constraint that xρx2 for all xS. Also, ϕ(xy)=ϕ(yx) so that xyρyx for all x,yS. A congruence ρ which satisfies these two conditions is called a semilattice congruence.

Conversely, a semilattice congruence ρ on S gives rise to a homomorphism from S to a semilattice S/ρ. The ρ-classes are the componentsMathworldPlanetmathPlanetmathPlanetmath of the decomposition.

Title semilattice decomposition of a semigroup
Canonical name SemilatticeDecompositionOfASemigroup
Date of creation 2013-03-22 13:07:09
Last modified on 2013-03-22 13:07:09
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Definition
Classification msc 20M10
Defines semilattice congruence