Universal specialization semilattices

TL;DR

This paper introduces a universal construction for specialization semilattices, embedding them into principal ones where closure is definable, thus unifying their structure across various scientific fields.

Contribution

It presents a canonical embedding of any specialization semilattice into a principal one where closure is explicitly definable, addressing a key limitation.

Findings

Every specialization semilattice can be embedded into a principal specialization semilattice.

Closure operation can be defined in the principal specialization semilattice.

The construction is canonical and universal.

Abstract

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.