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Line-closed combinatorial geometries

โœ Scribed by Mark D. Halsey

Publisher
Elsevier Science
Year
1987
Tongue
English
Weight
266 KB
Volume
65
Category
Article
ISSN
0012-365X

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โœฆ Synopsis

We present some results on combinatorial geometries (geometric lattices) in which closure is fine-closure. We prove that every interval of a line-closed geometry is line-closed. Furthermore, if every rank 3 interval of a geometry is line-closed, then the geometry is line-closed. This impfies that every supersolvable geometry is line-closed. Though not true in general, supersolvability does characterize the line-closed graphic geometries.

๐Ÿ“œ SIMILAR VOLUMES

Extensions of line-closed combinatorial
Extensions of line-closed combinatorial geometries
โœ Mark D. Halsey ๐Ÿ“‚ Article ๐Ÿ“… 1990 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 253 KB

In [4], line-closed combinatorial geometries were studied. Here, given a line-closed combinatorial geometry G(X), we determine all single point extensions of G(X) that are line-closed. Further, if H(X U r) is a line-closed geometry that is a smooth extension of G(X) we give a natural necessary and s

On the representations of projective geo
On the representations of projective geometries in algebraic combinatorial geometries
โœ Dฤ‚nuลฃ Marcu ๐Ÿ“‚ Article ๐Ÿ“… 1989 ๐Ÿ› Springer ๐ŸŒ English โš– 318 KB

In this paper, we show that the full algebraic combinatorial geometry is not a projective geometry, it is only semimodular, but the p-polynomial points give a projective subgeometry. Also, we show that the subgeometry can be coordinatized by a skew field, which is quotient ring of an Ore domain. As

Characterizing Combinatorial Geometries
Characterizing Combinatorial Geometries by Numerical Invariants
โœ Joseph E. Bonin; William P. Miller ๐Ÿ“‚ Article ๐Ÿ“… 1999 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 154 KB

We show that the projective geometry P G(r -1, q) for r > 3 is the only rank-r (combinatorial) geometry with (q r -1)/(q -1) points in which all lines have at least q + 1 points. For r = 3, these numerical invariants do not distinguish between projective planes of the same order, but they do disting