The bracket ring of a combinatorial geometry. I

Abstract:

The bracket ring is a ring of generalized determinants, called brackets, constructed on an arbitrary combinatorial geometry $G$. The brackets satisfy several familiar properties of determinants, including the syzygies, which are equivalent to Laplace’s expansion by minors. We prove that the bracket ring is a universal coordinatization object for $G$ in two senses. First, coordinatizations of $G$ correspond to homomorphisms of the ring into fields, thus reducing the study of coordinatizations of $G$ to the determination of the prime ideal structure of the bracket ring. Second, $G$ has a coordinatization-like representation over its own bracket ring, which allows an interesting generalization of some familiar results of linear algebra, including Cramer’s rule. An alternative form of the syzygies is then derived and applied to the problem of finding a standard form for any element of the bracket ring. Finally, we prove that several important relations between geometries, namely orthogonality, subgeometry, and contraction, are directly reflected in the structure of the bracket ring.