Intersection theorem
From Wikipedia, the free encyclopedia
In projective geometry, an intersection theorem or incidence theorem is an incidence structure consisting of points, lines, and possibly higher-dimensional objects and their incidences, together with a pair of nonincident objects A and B (for instance, a point and a line). The "theorem" states that, whenever a set of objects satisfies the incidences (i.e. can be matched up with the objects of the incidence structure in a way that preserves incidence), then the objects corresponding to A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is rather a property which some geometries satisfy but not others.
For example, Desargues' theorem can be stated using the following incidence structure:
- Points: {A,B,C,a,b,c,P,Q,R,O}
- Lines: {AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ}
- Incidences (in addition to obvious ones such as (A,AB): {(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)}
The implication is then (R,PQ)—that point R is incident with line PQ.
[edit] Famous examples
Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring D—
. The projective plane is then called desarguesian. A theorem of Amitsur's and Bergman's states that, in the context of desarguesian projective planes, for every intersection theorem there is a rational identity such that the plane P satisfies the intersection theorem if and only if the division ring D satisfies the rational identity.
- Pappus's hexagon theorem holds in a desarguesian projective plane
if and only if D is a field; it corresponds to the identity 
. - Fano's theorem (which states a certain intersection does not happen) holds in if and only if D has characteristic
; it corresponds to the identity a+a=0.
[edit] References
- L. H. Rowen; Polynomial Identities in Ring Theory. Academic Press: New York, 1980.
- S. A. Amitsur; "Rational Identities and Applications to Algebra and Geometry", Journal of Algebra 3 (1996), 304–359.
