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On the structure of modular lattices -- Unique gluing and dissection

Dale R. Worley

Abstract

This work proves that the process of gluing finite lattices to form a larger lattice is bijective, that is each lattice is the glued sum of a unique system of finite lattices, provided the class of lattices is constrained to modular, locally-finite lattices with finite covers. The results of this work are not surprising given the prior literature, but this seems to be the first proof that the processes of gluing and dissection can be made inverses, and hence that gluing is bijective.

On the structure of modular lattices -- Unique gluing and dissection

Abstract

This work proves that the process of gluing finite lattices to form a larger lattice is bijective, that is each lattice is the glued sum of a unique system of finite lattices, provided the class of lattices is constrained to modular, locally-finite lattices with finite covers. The results of this work are not surprising given the prior literature, but this seems to be the first proof that the processes of gluing and dissection can be made inverses, and hence that gluing is bijective.

Paper Structure

This paper contains 26 sections, 111 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions for definitions
  4. Finite covers
  5. Tolerances
  6. Properties of finite modular lattices
  7. Modular connected systems
  8. Gluing
  9. Basic properties of modular connected systems
  10. The sum
  11. The sum is a poset
  12. The 83240 intervals of S
  13. The relations of the 83233 intervals of L
  14. The sum is locally finite and has finite covers
  15. The sum is a lattice
  16. ...and 11 more sections

Key Result

Theorem \ref{th:inverses}

The two mappings (1) gluing a m.c. system to create a modular, l.f.f.c. lattice and (2) dissecting a modular, l.f.f.c. lattice to create a m.c. system are mutually inverse mappings between the category of isomorphism classes of m.c. systems and the category of isomorphism classes of modular, l.f.f.c

Theorems & Definitions (246)

  • Theorem \ref{th:inverses}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 236 more