## 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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