The lattice of semilattice-matrix decompositions of a semigroup

In this paper we investigate some general, lattice theoretical properties o f semilattice-matrix decompositions of semigroups. We prove that the poset of all semilattice-matrix equivalences on an arbitrary semigroup is a compl ete lattice. For a fixed semilattice congruence sigma on a semigroup S we p rove that the set of all semilattice-matrix equivalences on S carried by si gma is a complete sublattice of the lattice of equivalence relations on S, and that it is a direct product of the lattices of semilattice-left and sem ilattice-right equivalences on S carried by sigma.