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Derivable pseudo-nets. (English) Zbl 1171.51305

From the text: The main point of this note is to show that the two concepts of derivable nets and derivable pre pseudo-nets are completely equivalent. Our following main result establishes this result.
Theorem 2. Let \(D =(P,L,C,B)\) be an incidence structure of nonempty sets of ‘points’ \(P\), ‘lines’ \(L\) (each of which is incident with a nonempty set of points), nonempty sets of ‘parallel classes’ \(C\) and ‘Baer subplanes’ \(B\) (each of which is a non-degenerate affine plane) each with parallel class set \(C\) such that
(1) two distinct points are incident with at most one line,
(2) each line is incident with at least one point,
(3) each point is incident with a unique line of each parallel class, and
(4) each pair of distinct collinear points \(p\) and \(q\) are incident with a Baer subplane (non-degenerate affine plane) \(B(p, q)\).
Statements (1) through (4) simply state that \(D\) is a derivable pre pseudo-net. Then any two lines of distinct parallel classes intersect in a unique point and there are at least two points per line. A derivable pre pseudo-net is a (derivable) net.
We structure our proof as a series of lemmas.

MSC:

51E23 Spreads and packing problems in finite geometry
51A40 Translation planes and spreads in linear incidence geometry
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