The bracket ring of a combinatorial geometry. II. Unimodular geometries

Author:
Neil L. White

Journal:
Trans. Amer. Math. Soc. **214** (1975), 233-248

MSC:
Primary 05B35

DOI:
https://doi.org/10.1090/S0002-9947-1975-0447023-4

MathSciNet review:
0447023

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Abstract: The bracket ring of a combinatorial geometry *G* is a ring of generalized determinants which acts as a universal coordinatization object for *G*. Our main result is the characterization of a unimodular geometry as a binary geometry such that the radical of the bracket ring is a prime ideal. This implies that a unimodular geometry has a universal coordinatization over an integral domain, which domain we construct explicitly using multisets. An ideal closely related to the radical, the coordinatizing radical, is also defined and proved to be a prime ideal for every binary geometry.

To prove these results, we use two major preliminary theorems, which are of interest in their own right. The first is a bracket-theoretic version of Tutte's Homotopy Theorem for Matroids. We then prove that any two coordinatizations of a binary geometry over a given field are projectively equivalent.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0447023-4

Keywords:
Combinatorial geometry,
matroid,
bracket ring,
syzygy,
coordinatization,
binary geometry,
unimodular geometry,
projective equivalence,
coordinatizing prime ideal,
coordinatizing radical,
radical,
rank-preserving-weak-map image

Article copyright:
© Copyright 1975
American Mathematical Society