Veblen–Young theorem explained

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 (the set of points), together with a set L of subsets of P (the set of lines), 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 (meaning that there are two non-intersecting lines) then the projective space is isomorphic with the projective space of lines in a vector space over some division ring K.