- Intersection theorem
In
projective geometry , an intersection theorem or incidence theorem is anincidence 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 somedivision 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 arational 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.
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