On Brylawski's generalized duality

TL;DR

This paper extends Brylawski's duality concept to arbitrary rank functions, unifying various combinatorial structures and their polynomials, and characterizes matroids as objects that are both greedoids and dual greedoids.

Contribution

It introduces a generalized duality for rank functions, enabling new operations and polynomials, and characterizes matroids within this broader framework.

Findings

Generalized duality applies to greedoids, antimatroids, and demi-matroids.

A new polynomial satisfying deletion-contraction recursion is defined.

Matroids are characterized as objects that are both greedoids and dual greedoids.

Abstract

We introduce a notion of duality (due to Brylawski) that generalizes matroid duality to arbitrary rank functions. This generalized duality allows for generalized operations (deletion and contraction) and a generalized polynomial based on the matroid Tutte polynomial. This polynomial satisfies a deletion-contraction recursion. We explore this notion of duality for greedoids, antimatroids and demi-matroids, proving that matroids correspond precisely to objects that are simultaneously greedoids and "dual" greedoids.