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Transactions of the American Mathematical Society

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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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  • [1] T. H. Brylawski, A note on Tutte's unimodular representation theorem, Proc. Amer. Math. Soc. 52 (1975), 499-502. MR 0419271 (54:7294)
  • [2] T. H. Brylawski and D. Lucas, Uniquely representable combinatorial geometries, Internat. Colloq. on Combinatorial Theory, Rome, September, 1973.
  • [3] P. Camion, Modules unimodularies, J. Combinatorial Theory 4 (1968), 301-362. MR 48 #5918. MR 0327576 (48:5918)
  • [4] H. Crapo and G.-C. Rota, On the foundations of combinatorial theory: Combinatorial geometries, M. I. T. Press, Cambridge, Mass., 1970 (preliminary edition). MR 45 #74. MR 0290980 (45:74)
  • [5] D. Lucas, Properties of rank-preserving weak maps, Bull. Amer. Math. Soc. 80 (1974), 127-131. MR 0337665 (49:2434)
  • [6] -, Weak maps of combinatorial geometries, Trans. Amer. Math. Soc. 206 (1975), 247-279. MR 0371693 (51:7911)
  • [7] W. T. Tutte, A homotopy theorem for matroids. I, II, Trans. Amer. Math. Soc. 88 (1958), 144-174. MR 21 #336. MR 0101526 (21:336)
  • [8] P. Vámos, A necessary and sufficient condition for a matroid to be linear, Möbius Algebras (Proc. Conf., Univ. Waterloo, Waterloo, Ont. 1971), pp. 162-169. MR 0349447 (50:1941)
  • [9] N. L. White, The bracket ring and combinatorial geometry, Thesis, Harvard University, 1971.
  • [10] -, The bracket ring of a combinatorial geometry. I, Trans. Amer. Math. Soc. 202 (1975), 79-95. MR 0387095 (52:7942)
  • [11] O. Zariski and P. Samuel, Commutative algebra. Vols. I, II, University Ser. in Higher Math., Van Nostrand, Princeton, N. J., 1957, 1960. MR 19, 833; 22 #11006. MR 0120249 (22:11006)

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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

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