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Isomorphisms between Covering-Induced Lattices and Classical Geometric Lattices

Elvis Cabrera, Jyrko Correa

TL;DR

The paper studies when a lattice induced by a covering is isomorphic to canonical geometric lattices (partitions, subspaces, Dowling) by examining rank structure, closure operations, and the covering-atom relations in L(C). The authors develop a unified methodology based on the transversal matroid of a covering and derive necessary and sufficient conditions for isomorphisms, including equivalence reductions and uniformity considerations. They provide complete classifications for the isomorphism with partition lattices, subspace lattices, and Dowling lattices, clarifying the structural constraints under which these correspondences can occur. The results contribute to lattice theory and matroid theory by linking covering-induced lattices to well-known geometric lattices and informing potential applications in combinatorics.

Abstract

Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural properties of such lattices, with a particular focus on their rank structure, covering relations, and enumeration of elements per level. Leveraging these structural insights, we investigate necessary and sufficient conditions under which the lattice induced by a covering is isomorphic to classical geometric lattices, including the lattice of partitions, the lattice of subspaces of a vector space over a finite field, and the Dowling lattice. Our results provide a unified framework for comparing these combinatorial structures and contribute to the broader study of lattice theory, matroids, and their applications in combinatorics.

Isomorphisms between Covering-Induced Lattices and Classical Geometric Lattices

TL;DR

The paper studies when a lattice induced by a covering is isomorphic to canonical geometric lattices (partitions, subspaces, Dowling) by examining rank structure, closure operations, and the covering-atom relations in L(C). The authors develop a unified methodology based on the transversal matroid of a covering and derive necessary and sufficient conditions for isomorphisms, including equivalence reductions and uniformity considerations. They provide complete classifications for the isomorphism with partition lattices, subspace lattices, and Dowling lattices, clarifying the structural constraints under which these correspondences can occur. The results contribute to lattice theory and matroid theory by linking covering-induced lattices to well-known geometric lattices and informing potential applications in combinatorics.

Abstract

Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural properties of such lattices, with a particular focus on their rank structure, covering relations, and enumeration of elements per level. Leveraging these structural insights, we investigate necessary and sufficient conditions under which the lattice induced by a covering is isomorphic to classical geometric lattices, including the lattice of partitions, the lattice of subspaces of a vector space over a finite field, and the Dowling lattice. Our results provide a unified framework for comparing these combinatorial structures and contribute to the broader study of lattice theory, matroids, and their applications in combinatorics.
Paper Structure (15 sections, 19 theorems, 72 equations)

This paper contains 15 sections, 19 theorems, 72 equations.

Table of Contents

  1. Abstract
  2. Introduction
  3. Preliminaries
  4. The Matroid induced by a Covering
  5. Lattices
  6. Closed-Set Lattice of a Matroid
  7. Lattice Induced by a Covering
  8. The Lattice of Partitions
  9. The Lattice of Subspaces of a Vector Space over a Finite Field
  10. The Dowling Lattice
  11. Isomorphisms Between Lattices Induced by Coverings and Other Geometric Lattices
  12. Isomorphism with a Uniform Lattice
  13. Isomorphisms with the Lattice of Partitions
  14. Isomorphisms with subspace lattices
  15. Isomorphisms with the Dowling Lattice

Key Result

Proposition 3.3

Given a family of subsets $\mathcal{F} = \{F_i : i \in J\}$ of $U$, the pair $M(\mathcal{F}) = (U, \mathcal{J}(\mathcal{F}))$ defines a matroid, where $\mathcal{J}(\mathcal{F})$ consists of all partial transversals of $\mathcal{F}$. This matroid is called the transversal matroid associated with $\ma

Theorems & Definitions (38)

  • Definition 3.1: Matroid Whitney1935
  • Definition 3.2: Transversal Mirsky1971
  • Proposition 3.3: Transversal Matroid Mirsky1971
  • Theorem 3.4
  • Theorem 3.5
  • Definition 3.6: $G$-labeled blocks and partial $G$-partitions
  • Definition 3.7: Order and rank
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • ...and 28 more