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Nets satisfying the quadrangle condition. (English) Zbl 0637.51007
Summary: In the present work nets satisfying the following condition of closedness of quadrangles are investigated: If any four points of a net no three of them lying on a line can be joined by five lines, there exists a uniquely defined sixth line which also joins these points. The nets satisfying such a condition are called Q-nets. The work consists of four parts. The first part contains basic definitions and theorems. In the second part it is proved that every Q-net is an Ostrom net and every Ostrom net is a Q- net. In the third part there are studied some stronger closedness conditions which follow from the quadrangle condition, such that some parallelism of sides or some sides and diagonals are needed. In the fourth part it is proved that any Ostrom net over a Galois field can be embedded into a desarguesian plane. Further a classification of quadrangles in Ostrom nets and a formula for the number of such quadrangles are presented.
51E30 Other finite incidence structures (geometric aspects)