  A specialization semilattice is a join semilattice together with a coarser
preorder $ \sqsubseteq $ satisfying an appropriate compatibility condition. If
$X$ is a topological space, then $(\mathcal P(X), \cup, \sqsubseteq )$ is a
specialization semilattice, where $ x \sqsubseteq 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.
