Varieties of combinatorial geometries
HTML articles powered by AMS MathViewer
- by J. Kahn and J. P. S. Kung
- Trans. Amer. Math. Soc. 271 (1982), 485-499
- DOI: https://doi.org/10.1090/S0002-9947-1982-0654846-9
- PDF | Request permission
Abstract:
A hereditary class of (finite combinatorial) geometries is a collection of geometries which is closed under taking minors and direct sums. A sequence of universal models for a hereditary class $\mathcal {J}$ of geometries is a sequence $({T_n})$ of geometries in $\mathcal {J}$ with rank ${T_n} = n$, and satisfying the universal property: if $G$ is a geometry in $\mathcal {J}$ of rank $n$, then $G$ is a subgeometry of ${T_n}$. A variety of geometries is a hereditary class with a sequence of universal models. We prove that, apart from two degenerate cases, the only varieties of combinatorial geometries are (1) the variety of free geometries, (2) the variety of geometries coordinatizable over a fixed finite field, and (3) the variety of voltage-graphic geometries with voltages in a fixed finite group.References
- G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433-454.
—, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R.I., 1967.
- P. M. Cohn, Universal algebra, Harper & Row, Publishers, New York-London, 1965. MR 0175948
- 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
- Peter Doubilet, Gian-Carlo Rota, and Richard Stanley, On the foundations of combinatorial theory. VI. The idea of generating function, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 267–318. MR 0403987
- T. A. Dowling, A $q$-analog of the partition lattice, A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) North-Holland, Amsterdam, 1973, pp. 101–115. MR 0366707
- T. A. Dowling, A class of geometric lattices based on finite groups, J. Combinatorial Theory Ser. B 14 (1973), 61–86. MR 307951, DOI 10.1016/s0095-8956(73)80007-3 C. Greene, Lectures in combinatorial geometries, Notes from the NSF Seminar in Combinatorial Theory, Bowdoin College, 1971, unpublished.
- Saunders Mac Lane, A lattice formulation for transcendence degrees and $p$-bases, Duke Math. J. 4 (1938), no. 3, 455–468. MR 1546067, DOI 10.1215/S0012-7094-38-00438-7
- Douglas Kelly and Gian-Carlo Rota, Some problems in combinatorial geometry, A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) North-Holland, Amsterdam, 1973, pp. 309–312. MR 0357165
- Richard P. Stanley, Modular elements of geometric lattices, Algebra Universalis 1 (1971/72), 214–217. MR 295976, DOI 10.1007/BF02944981
- R. P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197–217. MR 309815, DOI 10.1007/BF02945028
- D. J. A. Welsh, Matroid theory, L. M. S. Monographs, No. 8, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0427112 T. Zaslavsky, Biased graphs, preprint, 1977.
- Thomas Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), no. 1, 47–74. MR 676405, DOI 10.1016/0166-218X(82)90033-6
Bibliographic Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 271 (1982), 485-499
- MSC: Primary 05B35; Secondary 51D20
- DOI: https://doi.org/10.1090/S0002-9947-1982-0654846-9
- MathSciNet review: 654846