  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.

Rings and Algebras;General Topology;Logic