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Diameter preserving bijections between Grassmann spaces over Bezout domains

Authors:	Li-Ping Huang

Institution:	(1) College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, 410076, China

Abstract:	Let R, S be Bezout domains. Assume that n is an integer ≥ 3, 1 ≤ k ≤ n − 2. Denoted by ${\mathbb{G}_k(_RR^n)}$ the k-dimensional Grassmann space on ${_RR^n}$ . Let ${\varphi: \mathbb{G}_k(_RR^n)\rightarrow \mathbb{G}_k(_SS^n)}$ be a map. This paper proves the following are equivalent: (i) ${\varphi}$ is an adjacency preserving bijection in both directions. (ii) ${\varphi}$ is a diameter preserving bijection in both directions. Moreover, Chow’s theorem on Grassmann spaces over division rings is extended to the case of Bezout domains: If ${\varphi: \mathbb{G}_k(_RR^n)\rightarrow \mathbb{G}_k(_SS^n)}$ is an adjacency preserving bijection in both directions, then ${\varphi}$ is induced by either a collineation or the duality of a collineation. Project 10671026 supported by National Natural Science Foundation of China.

Keywords:	Grassmann space Bezout domain Adjacency Diameter Preserving Projective space Collineation
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