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 $(\mathcal{P}(X),\cup ,⊑)$ is a specialization semilattice, where $x⊑y$ if $x\subseteq Ky$, for $x,y\subseteq X$, and $K$ is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. For short, the notion is useful since it allows us to consider a relation of “being generated by” with no need to require the existence of an actual “closure” or “hull”, which might be problematic in certain contexts. 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. We notice that a categorical argument guarantees the existence of universal embeddings in many parallel situations.