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