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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Tom Brylawski,*A note on Tutte’s unimodular representation theorem*, Proc. Amer. Math. Soc.**52**(1975), 499–502. MR**0419271**, https://doi.org/10.1090/S0002-9939-1975-0419271-6**[2]**T. H. Brylawski and D. Lucas,*Uniquely representable combinatorial geometries*, Internat. Colloq. on Combinatorial Theory, Rome, September, 1973.**[3]**P. Camion,*Modules unimodulaires*, J. Combinatorial Theory**4**(1968), 301–362 (French). MR**0327576****[4]**Henry H. Crapo and Gian-Carlo Rota,*On the foundations of combinatorial theory: Combinatorial geometries*, Preliminary edition, The M.I.T. Press, Cambridge, Mass.-London, 1970. MR**0290980****[5]**Dean Lucas,*Properties of rank preserving weak maps*, Bull. Amer. Math. Soc.**80**(1974), 127–131. MR**0337665**, https://doi.org/10.1090/S0002-9904-1974-13386-0**[6]**Dean Lucas,*Weak maps of combinatorial geometries*, Trans. Amer. Math. Soc.**206**(1975), 247–279. MR**0371693**, https://doi.org/10.1090/S0002-9947-1975-0371693-2**[7]**W. T. Tutte,*A homotopy theorem for matroids. I, II*, Trans. Amer. Math. Soc.**88**(1958), 144–174. MR**0101526**, https://doi.org/10.1090/S0002-9947-1958-0101526-0**[8]**P. Vámos,*A necessary and sufficient condition for a matroid to be linear*, Möbius algebras (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1971), Univ. Waterloo, Waterloo, Ont., 1971, pp. 162–169. MR**0349447****[9]**N. L. White,*The bracket ring and combinatorial geometry*, Thesis, Harvard University, 1971.**[10]**Neil L. White,*The bracket ring of a combinatorial geometry. I*, Trans. Amer. Math. Soc.**202**(1975), 79–95. MR**0387095**, https://doi.org/10.1090/S0002-9947-1975-0387095-9**[11]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR**0120249**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
05B35

Retrieve articles in all journals with MSC: 05B35

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