Weak cuts of combinatorial geometries
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- by Hien Q. Nguyen PDF
- Trans. Amer. Math. Soc. 250 (1979), 247-262 Request permission
Abstract:
A weak cut of a Combinatorial Geometry G is a generalization of a modular cut, corresponding to the family of the new dependent sets in a weak map image of G. The use of weak cuts allows the construction of all weak images of G, an important result being that, to any family ${\mathcal {M}}$ of independent sets of G, is associated a unique weak cut ${\mathcal {C}}$ containing ${\mathcal {M}}$. In practice, the flats of the weak image defined by ${\mathcal {C}}$ can be constructed directly. The weak cuts corresponding to known weak maps, such as truncation, projection, elementary quotient, are determined. The notion of weak cut is particularly useful in the study of erections. Given a geometry F and a weak image G, an F-erection of G is an erection of G which is a weak image of F. The main results are that the set of all F-erections of G is a lattice with the weak map order, and that the free F-erection can be constructed explicitly. Finally, a problem involving higher order erection is solved.References
- Henry H. Crapo, Single-element extensions of matroids, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 55–65. MR 190045, DOI 10.6028/jres.069B.003
- Henry H. Crapo, Erecting geometries, Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970) Univ. North Carolina, Chapel Hill, N.C., 1970, pp. 74–99. MR 0272655
- 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
- Thomas A. Dowling and Douglas G. Kelly, Elementary strong maps between combinatorial geometries, Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973) Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Rome, 1976, pp. 121–152 (English, with Italian summary). MR 0543658 D. A. Higgs, A lattice order on the set of all matroids on a set, Canad. Math. Bull. 9 (1966), 684-685.
- D. A. Higgs, Strong maps of geometries, J. Combinatorial Theory 5 (1968), 185–191. MR 231761, DOI 10.1016/S0021-9800(68)80054-7
- Donald E. Knuth, Random matroids, Discrete Math. 12 (1975), no. 4, 341–358. MR 406837, DOI 10.1016/0012-365X(75)90075-8
- Michel Las Vergnas, On certain constructions for matroids, Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975) Congressus Numerantium, No. XV, Utilitas Math., Winnipeg, Man., 1976, pp. 395–404. MR 0416956
- Dean Lucas, Weak maps of combinatorial geometries, Trans. Amer. Math. Soc. 206 (1975), 247–279. MR 371693, DOI 10.1090/S0002-9947-1975-0371693-2 H. Q. Nguyen, Constructing the free erection of a combinatorial geometry, J. Combinatorial Theory Ser. B (to appear).
- Hien Q. Nguyen, Projections and weak maps in combinatorial geometries, Discrete Math. 24 (1978), no. 3, 281–289. MR 523318, DOI 10.1016/0012-365X(78)90099-7
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 250 (1979), 247-262
- MSC: Primary 05B35
- DOI: https://doi.org/10.1090/S0002-9947-1979-0530054-7
- MathSciNet review: 530054