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 In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...

Non-Desarguesian plane In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective ...

s 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 Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life an ...

generalized the Veblen–Young theorem to

Tits building In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. B ...

s, showing that those of rank at least 3 arise from

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

s. 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 In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the elemen ...

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

''K''.

References

* * * * * {{DEFAULTSORT:Veblen-Young theorem Theorems in projective geometry Theorems in algebraic geometry