The Birkhoff completion of finite lattices

TL;DR

This paper introduces the Birkhoff completion for finite lattices, establishing its properties and applications, especially in ordinal data science, by embedding finite lattices into distributive lattices.

Contribution

It defines the Birkhoff completion as the minimal distributive lattice embedding for finite lattices and explores its connections to implicational theories and practical data analysis.

Findings

Birkhoff completion is the smallest distributive lattice containing a finite lattice.

It relates to implicational theories, especially R. Wille's simply-implicational theories.

Demonstrates application of Birkhoff completion in ordinal data science.

Abstract

We introduce the Birkhoff completion as the smallest distributive lattice in which a given finite lattice can be embedded as semi-lattice. We discuss its relationship to implicational theories, in particular to R. Wille's simply-implicational theories. By an example, we show how the Birkhoff completion can be used as a tool for ordinal data science.