Universal specialization semilattices

A specialization semilattice is a structure which can be embedded into $(\mathcal{P}(X),\cup ,⊑)$, where $X$ is a topological space, $x⊑y$ 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.