Universal extensions of specialization semilattices

A specialization semilattice is a join together with coarser preorder $ \sqsubseteq satisfying an appropriate compatibility condition. If $X$ topological space, then $(\mathcal P(X), \cup, )$ semilattice, where x y$ if $x \subseteq Ky$, for $x,y X$, and $K$ closure. Specialization semilattices posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. For short, the notion useful since it allows us consider relation of "being generated by" no need require existence actual "closure" or "hull", which might be problematic certain contexts. In former work we showed that every can embedded into associated space above. Here describe universal embedding additive closure semilattice. We notice categorical argument guarantees embeddings parallel situations.