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Net (in finite geometry)

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A net is an incidence structure of points and lines, i.e. a triple $(P,L,I)$ consisting of a set of points $P$, a set of lines $L$ and an incidence relation $I\subset P\times L$, such that

1) there exist points, $p$, and lines, $l$, and to every point (line) there exist two lines (points) not incident with it;

2) there is at most one line through every two distinct points;

3) if $p$ is not incident with $l$, then there exist one and only one line $m$ incident with $p$ that does not intersect $l$.

Thus, a net is a partial plane satisfying the parallelism axiom.

A more elaborate structure in finite geometry also goes by the name net, or geometrical net. It consists of a set of $n^2$ elements and three families of $n$ lines, with each line containing precisely $n$ points. This structure of lines and points is required to satisfy:

a) any two lines of two different families have precisely one point in common;

b) two lines from the same family have no point in common;

c) for each point there is exactly one line from each family passing through that point.

This notion relates to that of a web in differential geometry (cf. Webs, geometry of). Pictorially a geometrical net can be seen as an $(n\times n)$-array of points (squares). The first two families of lines form the columns and the rows, and the lines of the third family are subsets with precisely one point in each row and in each column. Giving the squares belonging to the $i$-th line of the third family the number $i$, $i=1,\ldots,n$, results in a Latin square (and vice versa).

References

[a1] R. Dembowski, "Finite geometries" , Springer (1968) pp. 254
[a2] F. Kårteszi, "Introduction to finite geometries" , North-Holland (1976)
[a3] L.M. Batten, "Combinatorics of finite geometries" , Cambridge Univ. Press (1986)
How to Cite This Entry:
Net (in finite geometry). Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Net_(in_finite_geometry)&oldid=37006