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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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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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