A combinatorial generalization of polytopes

Primoids and duoids are collections of subsets of a fixed finite set with a natural generalization of a pivoting property of convex polytopes. This structure is precisely what is necessary for the application of complementary pivoting algorithms. This paper investigates the combinatorial structure of primoids and duoids, showing them to form the circuits and cocircuits of a binary matroid. This matroid is then compared with the simplicial geometries of Crapo and Rota.