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A new proof of Funayama's theorem

Guram Bezhanishvili, Wesley H. Holliday

Abstract

Funayama proved that a lattice embeds into a complete Boolean algebra in such a way that all existing joins and meets are preserved if and only if the lattice satisfies the join-infinite and meet-infinite distributive laws. There are several proofs of this classic result in the literature. In this note, we provide a new and purely order-theoretic proof of Funayama's theorem, as well as of generalizations of the theorem.

A new proof of Funayama's theorem

Abstract

Funayama proved that a lattice embeds into a complete Boolean algebra in such a way that all existing joins and meets are preserved if and only if the lattice satisfies the join-infinite and meet-infinite distributive laws. There are several proofs of this classic result in the literature. In this note, we provide a new and purely order-theoretic proof of Funayama's theorem, as well as of generalizations of the theorem.
Paper Structure (3 sections, 9 theorems, 20 equations, 2 figures)

This paper contains 3 sections, 9 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Construction
  3. Connection to MacNeille completion

Key Result

Theorem 1.2

For any lattice $L$, there is a complete embedding of $L$ into a complete Boolean algebra iff $L$ satisfies JID and MID.

Figures (2)

  • Figure 1: the lattice $M_3$ (left) and $X_{M_3}$ (right). An arrow from $(x,y)$ to $(x',y')$ indicates that $(x',y')\sqsubseteq (x,y)$. The color of the nodes indicates the embedding $e: M_3\to \mathbf{B}M_3$.
  • Figure 2: the lattice $N_5$ (left) and $X_{N_5}$ (right). An arrow from $(x,y)$ to $(x',y')$ indicates that $(x',y')\sqsubseteq (x,y)$. The color of the nodes indicates the embedding $e:N_5\to \mathbf{B}N_5$, where purple pairs are in the $e$-image of both $a$ and $b$.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2: Funayama 1959
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2: Funayama for meet semilattices
  • proof
  • Remark 2.3
  • Theorem 2.4: Funayama for join semilattices
  • proof
  • Remark 2.5
  • ...and 12 more