Intersection theorem

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".

Famous examples

Desargues' theorem holds in a projective plane "P" if and only if "P" is the projective plane over some division ring "D"—$P=mathbb\{P\}\_\{2\}D$. 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 $mathbb\{P\}\_\{2\}D$ if and only if "D" is a field; it corresponds to the identity $forall\; a,bin\; D,acdot\; b=bcdot\; a$.
*Fano's theorem (which states a certain intersection does "not" happen) holds in $mathbb\{P\}\_\{2\}D$ if and only if "D" has characteristic $eq2$; it corresponds to the identity "a"+"a"=0.

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 no. 3 (1966), 304–359.