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 and (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 and 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 is incident with line .

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 and (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 and 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 is incident with line . Desargues' theorem holds in a projective plane if and only if is the projective plane over some division ring (skewfield) — 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 satisfies the intersection theorem if and only if the division ring satisfies the rational identity. Pappus's hexagon theorem holds in a desarguesian projective plane \mathbb{P}_{2}D if and only if 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 has characteristic \neq 2 ; it corresponds to the identity .

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