Intersection theorem
In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of 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 identified with the objects of the incidence structure in such a way that incidence is preserved), then the objects A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't.
For example, Desargues' theorem can be stated using the following incidence structure:
- Points: [math]\displaystyle{ \{A,B,C,a,b,c,P,Q,R,O\} }[/math]
- Lines: [math]\displaystyle{ \{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\} }[/math]
- Incidences (in addition to obvious ones such as [math]\displaystyle{ (A,AB) }[/math]): [math]\displaystyle{ \{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\} }[/math]
The implication is then [math]\displaystyle{ (R,PQ) }[/math]—that point R is incident with line PQ.
Famous examples
Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring (skewfield} D — [math]\displaystyle{ P=\mathbb{P}_{2}D }[/math]. The projective plane is then called desarguesian. A theorem of Amitsur and Bergman 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 [math]\displaystyle{ \mathbb{P}_{2}D }[/math] if and only if D is a field; it corresponds to the identity [math]\displaystyle{ \forall a,b\in D, \quad a\cdot b=b\cdot a }[/math].
- Fano's axiom (which states a certain intersection does not happen) holds in [math]\displaystyle{ \mathbb{P}_{2}D }[/math] if and only if D has characteristic [math]\displaystyle{ \neq 2 }[/math]; it corresponds to the identity a + a = 0.
References
- Rowen, Louis Halle, ed (1980). Polynomial Identities in Ring Theory. Pure and Applied Mathematics. 84. Academic Press. doi:10.1016/s0079-8169(08)x6032-5. ISBN 9780125998505.
- Amitsur, S. A. (1966). "Rational Identities and Applications to Algebra and Geometry". Journal of Algebra 3 (3): 304–359. doi:10.1016/0021-8693(66)90004-4.
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