Orlik–Solomon Type Algebras

We introduce χ -algebras, and show that a χ -algebra has the NBC basis property. We also show that a certain ideal used in the construction has the so-called BC basis property. The Orlik–Solomon algebra of a matroid, the Orlik–Terao algebra of a set of vectors, and the Cordovil algebra of an oriented matroid areχ -algebras. We define a new χ -algebra from a set of vectors, close to the Orlik–Terao, Orlik–Terao, and Cordovil algebras, but nevertheless different. Our proof provides a unified short and elementary proof of the NBC basis property for these algebras.