### Related pages

Subplanes of P(K,T)
Subplanes of PG(V)
Baer subplanes
Flats of a linear spaces
An incidence structure is a substructure of another incidence structure if , and We will write if is a substructure of .

In most cases of interest, the substructure is already determined by it's set of points. For example, if the linear space is a substructure of another linear space , then

Conversely, given any set of points, this set of lines yields a linear substructure.

Another important example are substructures of projective planes. A projective plane is called a subplane of another projective plane if , i.e. if is a substructure of . Of course, if , then the subsets and determine a subplane of if and only if for each two distinct points for each two distinct lines and if contains a quadrangle.

Primary examples are the fix structures of collineations, but these are possibly degenerated, i.e. usually there is no quadrangle fixed.

Contributed by Hauke Klein
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