Varieties of combinatorial geometries

Authors:
J. Kahn and J. P. S. Kung

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

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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 of geometries is a sequence of geometries in with rank , and satisfying the universal property: if is a geometry in of rank , then is a subgeometry of . 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.

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0654846-9

Article copyright:
© Copyright 1982
American Mathematical Society