universal extensions of specialization semilattices

A specialization semilattice is a join semilattice together with a coarser preorder ⊑ satisfying an appropriate compatibility condition. if x is a topological space, then (p(x),∪,⊑) is a specialization semilattice, where x ⊑ y if x ⊆ ky, for x, y ⊆ x, and k is closure. specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. in a former work we showed that every specialization semilattice can be embedded into the specialization semilattice associated to a topological space as above. here we describe the universal embedding of a specialization semilattice into an additive closure semilattice.