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Orlik-Solomon type algebras. (English) Zbl 0984.52016
Summary: We introduce $$\chi$$-algebras, and show that a $$\chi$$-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 are $$\chi$$-algebras. We define a new $$\chi$$-algebra from a set of vectors, close to the 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.

##### MSC:
 52C40 Oriented matroids in discrete geometry 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
##### Keywords:
no broken circuit; Orlik-Solomon algebra; matroid
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##### References:
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