[Paper Review] New representations of matroids and generalizations

Zur Izhakian, John Rhodes|arXiv (Cornell University)|Jan 1, 2011

Constraint Satisfaction and Optimization24 references15 citations

TL;DR

This paper introduces new representations of matroids and hereditary collections using matrices over finite semirings—specifically the boolean and superboolean semirings—extending classical field-based representations. It proves that any matroid decomposable into field-representable components admits a boolean representation, and all hereditary collections are superboolean-representable, generalizing classical representability theory.

ABSTRACT

We extend the notion of matroid representations by matrices over fields by considering new representations of matroids by matrices over finite semirings, more precisely over the boolean and the superboolean semirings. This idea of representations is naturally generalized to include hereditary collections (also known as abstract simplicial complexes). We show that a matroid that can be directly decomposed as matroids, each of which is representable over a field, has a boolean representation, and more generally that any arbitrary hereditary collection is superboolean-representable.

Motivation & Objective

• To generalize classical matroid representations over fields by introducing representations over finite semirings, particularly the boolean and superboolean semirings.
• To extend the concept of representability from matroids to broader combinatorial structures, including hereditary collections (abstract simplicial complexes).
• To establish conditions under which a matroid or hereditary collection can be represented via matrix structures over these semirings.
• To investigate the relationship between direct decomposition of matroids into field-representable components and their boolean representability.

Proposed method

• Defining matrix representations of matroids using operations in the boolean semiring (where addition is logical OR and multiplication is logical AND).
• Extending the framework to the superboolean semiring, which allows for a more refined algebraic structure while preserving combinatorial properties.
• Formulating a correspondence between the independent sets of a matroid and the row spaces of matrices over these semirings.
• Using the notion of direct decomposition of matroids into field-representable components to construct a boolean matrix representation.
• Proving that any hereditary collection (abstract simplicial complex) admits a superboolean matrix representation through structural decomposition and closure properties.
• Applying algebraic techniques from semiring theory to establish representability conditions and closure under hereditary operations.

Experimental results

Research questions

• RQ1 Can matroids that are direct sums of field-representable matroids be represented using matrices over the boolean semiring?
• RQ2 What is the relationship between the structure of a hereditary collection and its representability over the superboolean semiring?
• RQ3 How do semiring-based representations generalize classical field-based matroid representations?
• RQ4 Under what conditions does a matroid admit a boolean representation, particularly when decomposed into field-representable parts?
• RQ5 Can all abstract simplicial complexes (hereditary collections) be represented via matrices over the superboolean semiring?

Key findings

• Any matroid that is a direct sum of matroids representable over a field admits a boolean matrix representation.
• All hereditary collections, including abstract simplicial complexes, are superboolean-representable via appropriate matrix constructions.
• The boolean representation framework captures the combinatorial independence structure of decomposable matroids through semiring operations.
• The superboolean semiring provides a universal representability framework for hereditary collections, generalizing field-based representations.
• The proposed representations preserve the hereditary property of the original structures, ensuring consistency with combinatorial closure axioms.
• The results demonstrate that semiring-based representations offer a broader and more flexible algebraic foundation for matroid and simplicial complex theory than classical field representations.