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On normal subnets of algebraic nets. (Russian) Zbl 0724.51004
This paper is a contribution to the theory of k-nets with arbitrary \(k\geq 3\), which are called “algebraic nets of arbitrary kind” by the author. Using the concept of homotopy previously introduced by V. D. Bělousov for \(k=3\), and generalized by the author himself for arbitrary \(k\geq 3\), the concept of a normal subset of a k-net is introduced and studied. In particular, on condition that \(N=({\mathcal P};{\mathcal L})\) and \(N'=({\mathcal P}';{\mathcal L}')\) are two k-nets of the same kind, and \({\bar \phi}\): \(N\to N'\) is a homotopy of N into \(N'\), then the following two main results are proved: (1) pre-image of any point of the net \(N'\) in the homotopy \({\bar \phi}\) is a normal subnet of N, and (2) any normal subset H of the net N corresponding to the homotopy \({\bar \phi}\) uniquely determines all normal subnets of N corresponding to \({\bar \phi}\). Using this the concept of a quotient net of the net N by its normal subnet H is introduced and its construction is described at the end of the paper.
Reviewer: J.Libicher (Brno)
MSC:
51E14 Finite partial geometries (general), nets, partial spreads
51A25 Algebraization in linear incidence geometry
Keywords:
k-nets; quotient net
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