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On the structure of modular lattices -- Axioms for gluing

Dale R. Worley

TL;DR

The paper advances lattice gluing by proposing two complementary axiom schemes for modular, locally finite lattices with finite covers: maximal axioms that maximize immediate inferable structure and minimal axioms that minimize assumptions while preserving equivalence to the baseline framework. It replaces and extends previous baselines with local-interval conditions (via $[x \wedge y, x \vee y]$) and a system of partial bijections (or omega maps) to connect blocks, showing that the minimal and maximal formulations are equivalent to the traditional baseline, and that extensions to full maps recover global structure such as the diamond isomorphism theorem in modular sums. The work includes concrete definitional devices (e.g., 83306, 83263) and dual axiom forms, enabling streamlined verification and classification of valid gluings. The results offer a compact, locally-specified toolkit for constructing and analyzing polytone and monotone gluing configurations in modular lattice theory, with potential implications for understanding global lattice features from local compatibilities.

Abstract

This paper explores alternative statements of the axioms for lattice gluing, focusing on lattices that are modular, locally finite, and have finite covers, but may have infinite height. We give a set of "maximal" axioms that maximize what can be immediately adduced about the structure of a valid gluing. We also give a set of "minimal" axioms that minimize what needs to be adduced to prove that a system of blocks is a valid gluing. This system appears to be novel in the literature. A distinctive feature of the minimal axioms is that they involve only relationships between elements of the skeleton which are within an interval $[x \wedge y, x \vee y]$ where either $x$ and $y$ cover $x \wedge y$ or they are covered by $x \vee y$. That is, they have a decidedly local scope, despite that the resulting sum lattice, being modular, has global structure, such as the diamond isomorphism theorem.

On the structure of modular lattices -- Axioms for gluing

TL;DR

The paper advances lattice gluing by proposing two complementary axiom schemes for modular, locally finite lattices with finite covers: maximal axioms that maximize immediate inferable structure and minimal axioms that minimize assumptions while preserving equivalence to the baseline framework. It replaces and extends previous baselines with local-interval conditions (via ) and a system of partial bijections (or omega maps) to connect blocks, showing that the minimal and maximal formulations are equivalent to the traditional baseline, and that extensions to full maps recover global structure such as the diamond isomorphism theorem in modular sums. The work includes concrete definitional devices (e.g., 83306, 83263) and dual axiom forms, enabling streamlined verification and classification of valid gluings. The results offer a compact, locally-specified toolkit for constructing and analyzing polytone and monotone gluing configurations in modular lattice theory, with potential implications for understanding global lattice features from local compatibilities.

Abstract

This paper explores alternative statements of the axioms for lattice gluing, focusing on lattices that are modular, locally finite, and have finite covers, but may have infinite height. We give a set of "maximal" axioms that maximize what can be immediately adduced about the structure of a valid gluing. We also give a set of "minimal" axioms that minimize what needs to be adduced to prove that a system of blocks is a valid gluing. This system appears to be novel in the literature. A distinctive feature of the minimal axioms is that they involve only relationships between elements of the skeleton which are within an interval where either and cover or they are covered by . That is, they have a decidedly local scope, despite that the resulting sum lattice, being modular, has global structure, such as the diamond isomorphism theorem.

Paper Structure

This paper contains 14 sections, 23 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattice properties
  4. Tolerances
  5. Ordered tolerances
  6. Partial bijections
  7. Baseline axioms
  8. Minimal axioms
  9. The axioms
  10. Defining 83306
  11. Strengthening (PS790400239042047) and (PS890400239042050)
  12. Defining 83263
  13. Equivalence
  14. Maximal axioms

Key Result

Lemma 2.4

Band1981a*Lem. 1.1 If $\mathrel{\xi}$ is a tolerance on a lattice $L$, then:

Figures (2)

  • Figure 1: Configuration in the proof of lem. \ref{['lem:chain-indep-meet']}
  • Figure 2: Configuration in the proof of lem. \ref{['lem:join-from-cover']}

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 44 more