Veblen–Young theorem


In mathematics, the Veblen–Young theorem, proved by, states that a projective space of dimension at least 3 can be constructed as the projective space associated to a vector space over a division ring.
Non-Desarguesian planes give examples of 2-dimensional projective spaces that do not arise from vector spaces over division rings, showing that the restriction to dimension at least 3 is necessary.
Jacques Tits generalized the Veblen–Young theorem to Tits buildings, showing that those of rank at least 3 arise from algebraic groups.
generalized the Veblen–Young theorem to continuous geometry, showing that a complemented modular lattice of order at least 4 is isomorphic to the principal right ideals of a von Neumann regular ring.

Statement

A projective space S can be defined abstractly as a set P, together with a set L of subsets of P, satisfying these axioms :
  • Each two distinct points p and q are in exactly one line.
  • Veblen's axiom: If a, b, c, d are distinct points and the lines through ab and cd meet, then so do the lines through ac and bd.
  • Any line has at least 3 points on it.
The Veblen–Young theorem states that if the dimension of a projective space is at least 3 then the projective space is isomorphic with the projective space of lines in a vector space over some division ring K.