Veblen–Young theorem
In mathematics, the Veblen–Young theorem, proved by {{harvs|txt|author1-link=Oswald Veblen|author2-link=John Wesley Young|first1=Oswald|last1=Veblen|first2=John Wesley|last2=Young|year1=1908|year2=1910|year3=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.
{{harvs|txt|authorlink=John von Neumann|first=John|last=von Neumann|year=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
- {{Citation | last1=Cameron | first1=Peter J. | title=Projective and polar spaces | url=http://www.maths.qmul.ac.uk/~pjc/pps/ | publisher=Queen Mary and Westfield College School of Mathematical Sciences | location=London | series=QMW Maths Notes | isbn=978-0-902480-12-4 | mr=1153019 | year=1992 | volume=13}}
- {{Citation | last1=Veblen | first1=Oswald | author1-link=Oswald Veblen | last2=Young | first2=John Wesley | title=A Set of Assumptions for Projective Geometry | doi=10.2307/2369956 | mr=1506049 | year=1908 | journal=American Journal of Mathematics | issn=0002-9327 | volume=30 | issue=4 | pages=347–380| jstor=2369956 }}
- {{Citation | last1=Veblen | first1=Oswald | author1-link=Oswald Veblen | last2=Young | first2=John Wesley | title=Projective geometry Volume I | url=https://archive.org/details/projectivegeome00veblgoog | publisher=Ginn and Co., Boston | isbn=978-1-4181-8285-4 | mr=0179666 | year=1910}}
- {{Citation | last1=Veblen | first1=Oswald | author1-link=Oswald Veblen | last2=Young | first2=John Wesley | title=Projective geometry Volume II | url=https://archive.org/details/projectivegeomet028875mbp | publisher=Ginn and Co., Boston | isbn=978-1-60386-062-8 | mr=0179667 | year=1917}}
- {{Citation | last1=von Neumann | first1=John | author1-link=John von Neumann | title=Continuous geometry | orig-year=1960 | url=https://books.google.com/books?id=onE5HncE-HgC | publisher=Princeton University Press | series=Princeton Landmarks in Mathematics | isbn=978-0-691-05893-1 | mr=0120174 | year=1998}}
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