Regular homomorphisms of generalized projective planes |
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Authors: | Frieder Knüppel |
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Institution: | (1) Mathematisches Seminar der Universität, Olshausenstrasse 40-60, 23 Kiel, W. Germany |
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Abstract: | A generalized projective plane is an incidence structure together with a relation distant on the set of points and also on the set of lines, such that any two distant points A,B (lines a,b) have a unique common line (A,B) (common point (a,b)) and three further axioms hold. Every commutative ring with 1 supplies a model. A homomorphism of into an incidence structure
is called regular if the following condition and its dual are valid: A distant B and c IA ,B implies c =(A,B) . We shall prove the following two theorems. Let be a generalized projective plane satisfying a richness condition called (U). Let M I m. If and are regular homomorphisms of such that X = M X = M for each point X of the line m then A = B A = B for any two points A,B. If is a projective plane over a commutative ring such that (U) holds then the surjective regular homomorphisms of are induced by the ideals of the ring; in particular, the image of under a regular homomorphism is again a projective plane over a ring, and preserves distant . |
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