Universal extensions of specialization semilattices

Abstract

. 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. For short, the notion is useful since it allows us to consider a relation of “being generated by” with no need to require the existence of an actual “closure” or “ hull”, which is problematic in certain contexts. 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. We notice that a categorical argument guarantees the existence of universal embeddings in many parallel situations.