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Adjacency preserving mappings. (English) Zbl 1027.51003

Two points in an incidence structure are called adjacent if there exists a line passing through them. The authors study maps between locally projective near \(2n\)-gons preserving adjacency. A near \(2n\)-gon \(G\) is called locally projective if the line pencil through each point has the structure of a projective space in a natural way, all of the same dimension \(m-1\geq 1\). In this case \(m\) is called the rank of \(G\).
The following theorem is proved: Let \(G\) and \(G'\) be locally projective near \(2n\)-gons (\(n\geq 1\)) of the same finite rank \(m\geq 2\). Then every surjective, adjacency preserving map on the point sets is an isomorphism of the geometries. This applies in particular to generalized \(2n\)-gons (\(n>1\)) and dual polar spaces.
The theorem generalizes earlier work of W. Huang [Proc. Am. Math. Soc. 128, 2451-2455 (2000; Zbl 0955.51004)]. The authors make a case that there is not much room to further generalize the result.

MSC:

51A50 Polar geometry, symplectic spaces, orthogonal spaces
51E12 Generalized quadrangles and generalized polygons in finite geometry

Citations:

Zbl 0955.51004
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References:

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