Intersection theorem

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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: $\{A,B,C,a,b,c,P,Q,R,O\}$
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 $PQ$ .

Famous examples

Desargues' theorem holds in a projective plane $P$ if and only if $P$ is the projective plane over some division ring (skewfield) $D$ — $P=\mathbb {P} _{2}D$ . 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 $\mathbb {P} _{2}D$ if and only if $D$ is a field; it corresponds to the identity $\forall a,b\in D,\quad a\cdot b=b\cdot a$ .
Fano's axiom (which states a certain intersection does not happen) holds in $\mathbb {P} _{2}D$ if and only if $D$ has characteristic $\neq 2$ ; it corresponds to the identity $a + a = 0$ .

References

Rowen, Louis Halle, ed. (1980). Polynomial Identities in Ring Theory. Pure and Applied Mathematics. Vol. 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.