A Distributive Lattice Cover for Semilattices

TL;DR

This paper explores two methods for constructing an envelope for finite locally distributive strong upper semilattices, demonstrating their equivalence through isomorphism.

Contribution

It introduces and compares two novel constructions of envelopes for finite locally distributive semilattices, establishing their isomorphism.

Findings

The two envelope constructions are isomorphic.

Birkhoff's representation and valuations produce equivalent envelopes.

The methods unify the understanding of semilattice extensions.

Abstract

We consider two constructions of an envelope for a finite locally distributive strong upper semilattice. The first is based on Birkhoff's representation of finite distributive lattices and the second on valuations on lattices. We show that these produce isomorphic envelopes.