Universal specialization semilattices

Abstract

A specialization semilattice is a structure which can be embedded into (P(X), ∪, ⊑), where X is a topological space, x ⊑ y means x ⊆ Ky, for  x, y ⊆ 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.