Veblen–Young theorem

In mathematics, the Veblen–Young theorem, proved by Oswald Veblen and John Wesley Young (1908, 1910, 1917), 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.

John von Neumann (1998) 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.

References

Cameron, Peter J. (1992), Projective and polar spaces, QMW Maths Notes, vol. 13, London: Queen Mary and Westfield College School of Mathematical Sciences, ISBN 978-0-902480-12-4, MR 1153019
Veblen, Oswald; Young, John Wesley (1908), "A Set of Assumptions for Projective Geometry", American Journal of Mathematics, 30 (4): 347–380, doi:10.2307/2369956, ISSN 0002-9327, JSTOR 2369956, MR 1506049
Veblen, Oswald; Young, John Wesley (1910), Projective geometry Volume I, Ginn and Co., Boston, ISBN 978-1-4181-8285-4, MR 0179666
Veblen, Oswald; Young, John Wesley (1917), Projective geometry Volume II, Ginn and Co., Boston, ISBN 978-1-60386-062-8, MR 0179667
von Neumann, John (1998) [1960], Continuous geometry, Princeton Landmarks in Mathematics, Princeton University Press, ISBN 978-0-691-05893-1, MR 0120174