Paolo Lipparini - Dipartimento di Matematica, Viale della Ricerca Scientifica Non Chiusa, Universit`a di Roma “Tor Vergata”, I-۰۰۱۳۳ Rome, Italy.

A specialization semilattice is a join semilattice together with a coarser preorder ⊑ satisfying an appropriate compatibility condition. If X is a topological space, then (P(X),∪,⊑) is a specialization semilattice, where x ⊑ y if x ⊆ Ky, for x, y ⊆ X, and K is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. In a former work we showed that every specialization semilattice can be embedded into the specialization semilattice associated to a topological space as above. Here we describe the universal embedding of a specialization semilattice into an additive closure semilattice.

Specialization semilattice, closure semilattice, closure space, universal extension