Universal extensions of specialization semilattices

TL;DR

This paper explores the universal embedding of specialization semilattices into additive closure semilattices, providing a categorical approach that broadens their applicability across various scientific fields.

Contribution

It introduces a universal embedding of specialization semilattices into additive closure semilattices, extending previous work and utilizing categorical methods for broader applicability.

Findings

Universal embedding into additive closure semilattices established

Categorical argument guarantees existence of universal embeddings in many contexts

Application potential across diverse scientific disciplines

Abstract

A specialization semilattice is a join semilattice together with a coarser preorder $ \sqsubseteq $ satisfying an appropriate compatibility condition. If $X$ is a topological space, then $(\mathcal P(X), \cup, \sqsubseteq )$ is a specialization semilattice, where $ x \sqsubseteq y$ if $x \subseteq Ky$, for $x,y \subseteq X$, 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 might be 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.