Universal specialization semilattices

A specialization semilattice is a structure which can be embedded into $(\mathcal P(X), \cup, \sqsubseteq )$, where $X$ is a topological space, $ x \sqsubseteq y$ means $x \subseteq Ky$, for $x,y \subseteq X$, and $K$ is closure in $X$. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. In general, closure is not expressible in a specialization semilattice. On the other hand, we show that every specialization semilattice can be canonically embedded into a "principal" specialization semilattice in which closure can be actually defined.Comment: Treats the nonadditive case; the additive case is somewhat simpler and has been treated in arXiv:2201.09083 v2: added a few detail