Preserving meets in meet-dense poset completions

TL;DR

This paper characterizes conditions under which a poset can be embedded into a complete lattice that preserves certain meets, revealing structural properties of these completions and their lattice-theoretic characteristics.

Contribution

It provides necessary and sufficient conditions for meet-preserving embeddings into completions and analyzes the lattice structure of these completions, especially for finite posets.

Findings

Set M of completions forms a topped weakly lower semimodular lattice

For finite P, M is a lower semimodular lattice

An example shows M may lack a bottom element

Abstract

Defining P* to be the complete lattice of upsets (ordered by reverse inclusion) of a poset P we give necessary and sufficient conditions on a subset S of P* for P to admit a meet-completion e from P to Q where e preserves the infimum of an upwardly closed set from P if and only if it is in S. We show that given S satisfying these conditions the set M of these completions forms a topped weakly lower semimodular lattice. In particular, when P is finite M is a lower semimodular lattice, and a lower bounded homomorphic image of a free lattice. We provide an example where M does not have a bottom element.