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....