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The Birkhoff completion of finite lattices

Mohammad Abdulla, Johannes Hirth, Gerd Stumme

TL;DR

The paper defines the Birkhoff completion $BC(L)$ as the minimal distributive lattice into which a finite lattice $L$ embeds as a join-subsemilattice, grounding the construction in Birkhoff's representation and its FCA manifestations. It then extends the construction to formal contexts, showing $BC(\mathbb{K})$ aligns with the concept lattice of $BC(\mathbb{K})$ and links to Wille's implicational theories through the canonical direct basis. An implicational approach demonstrates that $BC(\mathbb{K})$ can be computed by refining the basis to singleton premises and closing under the resulting theory, providing a practical computation pathway. The UK geographic example illustrates how $BC$ reveals complement-based concepts and potential new objects, offering a tool for ordinal data analysis. Overall, the work unifies lattice theory and FCA to obtain distributive completions with meaningful interpretation and applications in 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.

The Birkhoff completion of finite lattices

TL;DR

The paper defines the Birkhoff completion as the minimal distributive lattice into which a finite lattice embeds as a join-subsemilattice, grounding the construction in Birkhoff's representation and its FCA manifestations. It then extends the construction to formal contexts, showing aligns with the concept lattice of and links to Wille's implicational theories through the canonical direct basis. An implicational approach demonstrates that can be computed by refining the basis to singleton premises and closing under the resulting theory, providing a practical computation pathway. The UK geographic example illustrates how reveals complement-based concepts and potential new objects, offering a tool for ordinal data analysis. Overall, the work unifies lattice theory and FCA to obtain distributive completions with meaningful interpretation and applications in 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.
Paper Structure (7 sections, 10 theorems, 10 equations, 11 figures)

This paper contains 7 sections, 10 theorems, 10 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Basics
  3. Fixing non distributivity based on Birkhoff's Representation Theorem
  4. The Birkhoff completion of a formal context
  5. An implicational approach to the Birkhoff completion
  6. Application: analysis of the administrative geography of the UK
  7. Conclusion

Key Result

theorem 1

A lattice $L$ is distributive if and only if there exists no sublattice $A \subseteq L$ isomorphic to either $M_3$ or $N_5$.

Figures (11)

  • Figure 1: The nondistributive lattices $M_3$ and $N_5$
  • Figure 2: $M_3$ (left) , its down-set $BC$ (middle) and its up-set $BC$(Right).
  • Figure 3: $N_5$ (left), its down-set $BC$ (middle) and its up-set $BC$(Right).
  • Figure 4: A lattice for which there exists an order embedding into a distributive lattice that has fewer elements than the up-set or down-set Birkhoff completion.
  • Figure 5: $L$ (left), its down-set $BC$ (middle) and its up-set $BC$ (right).
  • ...and 6 more figures

Theorems & Definitions (21)

  • definition 1
  • theorem 1
  • theorem 2: Birkhoff's Representation Theorem
  • theorem 3: Dual version of Birkhoff's Representation Theorem
  • corollary 1
  • definition 2
  • remark 1
  • theorem 4
  • proof
  • lemma 1
  • ...and 11 more