teorema da interseção

Intersection theorem This article is about projective geometry. For a result on tensor products of modules, see Homological conjectures in commutative algebra.

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 (por exemplo, a point and a line). o "teorema" afirma que, whenever a set of objects satisfies the incidences (ou seja. 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.

Por exemplo, Desargues' theorem can be stated using the following incidence structure: Points: {estilo de exibição {UMA,B,C,uma,b,c,P,Q,R,O}} Lines: {estilo de exibição {AB,CA,BC,ab,ac,bc,Aa,Bb,Cc,PQ}} Incidences (in addition to obvious ones such as {estilo de exibição (UMA,AB)} ): {estilo de exibição {(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,CA),(Q,ac),(R,AB),(R,ab)}} The implication is then {estilo de exibição (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 — {displaystyle 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 {estilo de exibição mathbb {P} _{2}D} if and only if D is a field; it corresponds to the identity {displaystyle forall a,bin D,quad acdot b=bcdot a} . Fano's axiom (which states a certain intersection does not happen) holds in {estilo de exibição mathbb {P} _{2}D} if and only if D has characteristic {estilo de exibição neq 2} ; it corresponds to the identity a + a = 0. References Rowen, Louis Halle, ed. (1980). Polynomial Identities in Ring Theory. Matemática Pura e Aplicada. Volume. 84. Imprensa Acadêmica. doi:10.1016/s0079-8169(08)x6032-5. ISBN 9780125998505. Amitsur, S. UMA. (1966). "Rational Identities and Applications to Algebra and Geometry". Jornal de Álgebra. 3 (3): 304–359. doi:10.1016/0021-8693(66)90004-4. Categorias: Incidence geometryTheorems in projective geometry