[Paper Review] Boolean Representations of Matroids and Lattices

Zur Izhakian, John Rhodes|arXiv (Cornell University)|Aug 6, 2011

Topological and Geometric Data Analysis16 references11 citations

TL;DR

This paper introduces a novel boolean matrix representation for finite lattices, demonstrating that every matroid is boolean-representable by leveraging its associated lattice of flats. The method provides a constructive, computationally accessible framework that links matroid geometry with boolean algebra, yielding tighter bounds on representation size and extending to tropical and idempotent semiring representations.

ABSTRACT

We introduce a new representation concept for lattices by boolean matrices, and utilize it to prove that any matroid is boolean representable. We show that such a representation can be easily extracted from a representation of the associated lattice of flats of the matroid, leading also to a tighter bound on the representation's size. Consequently, we obtain a linkage of boolean representations with geometry in a very natural way.

Motivation & Objective

• To develop a new representation framework for finite lattices using boolean matrices.
• To establish a constructive link between matroid geometry and boolean algebra through the lattice of flats.
• To provide a tighter upper bound on the size of boolean representations of matroids.
• To generalize the results to tropical and idempotent semirings, showing broader representability.

Proposed method

• Defining c-independence and c-rank for lattices via boolean matrix representations, ensuring compatibility with lattice height.
• Constructing a boolean representation of a matroid by restricting the representation of its lattice of flats to the atoms of the lattice.
• Using the natural embedding of the boolean semiring into the tropical semiring to extend results to tropically representable matroids.
• Applying the concept of sup-generating subsets and their partitions to characterize c-dependence in lattices.
• Utilizing the Grassmann-Plücker map over the superboolean semiring as a foundation for boolean independence.
• Deriving an upper bound on representation size via the sum of binomial coefficients up to the matroid rank.

Experimental results

Research questions

• RQ1 Can every matroid be represented using boolean matrices, and if so, how can such a representation be systematically constructed?
• RQ2 How does the c-rank of a lattice’s boolean representation relate to its geometric height and chain structure?
• RQ3 What is the tightest possible upper bound on the size of a boolean representation for a given matroid?
• RQ4 To what extent can boolean representations be generalized to other algebraic structures like tropical or idempotent semirings?
• RQ5 How do the notions of c-independence and c-rank in lattices correspond to known combinatorial and geometric properties?

Key findings

• Every matroid is boolean-representable, as proven by constructing a boolean matrix representation from the lattice of flats of the matroid.
• The c-rank of the boolean representation of a lattice equals the height of the lattice, establishing a strong structural correspondence.
• The representation size of a matroid is bounded above by the sum of binomial coefficients ∑ᵢ₌₀ᵏ (ⁿᵢ), where n is the ground set size and k is the rank.
• The boolean representation can be extracted directly from the lattice of flats, providing a systematic and geometrically natural construction method.
• All matroids are tropically representable, and more generally, representable over any idempotent semiring, via the embedding of the boolean semiring into such structures.
• The method yields a tighter upper bound on representation size than previous approaches, particularly when considering strict join irreducibles.