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Simplifying modular lattices by removing doubly irreducible elements

Jukka Kohonen

TL;DR

The work introduces a structural reduction for modular lattices by removing selected doubly irreducible elements (trinkets) to form racks, enabling compact representations of the large family of unlabeled vertically indecomposable modular lattices ($\mathrm{MV}_n$). It establishes that racks preserve key properties and can be decorated to recover the full lattices, with ornamentation counted via Polya enumeration using cycle indices. The authors perform extensive computations up to $n\le 40$, producing a virtual listing of $\mathrm{MV}_n$ and an accompanying explicit count of all modular lattices, and provide SageMath tools for on-demand generation and sampling. This approach yields dramatic storage savings (over 3000x) and scalable access to large lattice families, while offering a pathway to further optimization through gluing and database-style querying. The work combines structural theory, exhaustive computation, and software to make large-scale lattice data more tractable and usable for researchers.

Abstract

Lattices are simplified by removing some of their doubly irreducible elements, resulting in smaller lattices called racks. All vertically indecomposable modular racks of $n \le 40$ elements are listed, and the numbers of all modular lattices of $n \le 40$ elements are obtained by Pólya counting. SageMath code is provided that allows easy access both to the listed racks, and to the modular lattices that were not listed. More than 3000-fold savings in storage space are demonstrated.

Simplifying modular lattices by removing doubly irreducible elements

TL;DR

The work introduces a structural reduction for modular lattices by removing selected doubly irreducible elements (trinkets) to form racks, enabling compact representations of the large family of unlabeled vertically indecomposable modular lattices (). It establishes that racks preserve key properties and can be decorated to recover the full lattices, with ornamentation counted via Polya enumeration using cycle indices. The authors perform extensive computations up to , producing a virtual listing of and an accompanying explicit count of all modular lattices, and provide SageMath tools for on-demand generation and sampling. This approach yields dramatic storage savings (over 3000x) and scalable access to large lattice families, while offering a pathway to further optimization through gluing and database-style querying. The work combines structural theory, exhaustive computation, and software to make large-scale lattice data more tractable and usable for researchers.

Abstract

Lattices are simplified by removing some of their doubly irreducible elements, resulting in smaller lattices called racks. All vertically indecomposable modular racks of elements are listed, and the numbers of all modular lattices of elements are obtained by Pólya counting. SageMath code is provided that allows easy access both to the listed racks, and to the modular lattices that were not listed. More than 3000-fold savings in storage space are demonstrated.
Paper Structure (8 sections, 8 theorems, 7 equations, 6 figures, 1 table)

This paper contains 8 sections, 8 theorems, 7 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and basic results
  3. Decoration under symmetry
  4. Generating and classifying the racks
  5. Numerical results
  6. SageMath implementation
  7. Related work
  8. Concluding remarks

Key Result

Lemma 2.6

$L$ and $\mathop{\mathrm{Rack}}\nolimits L$ have the same decoration sites, and $\mathop{\mathrm{Rack}}\nolimits L$ is a rack.

Figures (6)

  • Figure 1: A modular lattice and its rack.
  • Figure 2: All modular vi-racks of $1$ to $13$ elements, labeled with the number of elements and an ordinal index.
  • Figure 3: A modular rack and its seven nonisomorphic decorations with two trinkets.
  • Figure 4: A rack with eleven decoration sites.
  • Figure 5: Numbers of modular lattices, modular vi-lattices, modular vi-racks, distributive vi-lattices, and planar distributive vi-lattices of $n$ elements (all up to isomorphism).
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 13 more