nLab
quasifield

Outline

A quasifield (earlier also called a Veblen-Wedderburn system) is an algebraic structure with two binary operations ++ and \cdot which is weaker than a division ring and which is motivated by synthetic projective geometry. There are the left and right versions. Associative right quasifield is the same as a near-field.

Definition

A left quasifield is an algebraic structure (A,+,)(A,+,\cdot) with two binary operations ++ and \cdot such that

  • (A,+)(A,+) is a group with neutral element 00
  • (A\{0},)(A\backslash \{0\},\cdot) is a loop
  • left distributivity a(b+c)=ab+aca\cdot(b+c) = a\cdot b + a\cdot c for all a,b,cAa,b,c\in A
  • if a,b,cAa,b,c\in A, aba\neq b then the equation ax=bx+ca\cdot x = b \cdot x + c has a unique solution for xAx\in A

A left quasifield is Abelian if the underlying group (A,+)(A,+) is Abelian.

Literature

  • wikipedia quasifield
  • O. Veblen, J.H.M. Wedderburn, Non-Desarguesian and non-Pascalian geometries, Trans AMS 8, 379–388 (1907) pdf
  • Marshall Hall Jr., Projective planes, Trans. Amer. Math. Soc. 54: 229–277 (1943) pdf
  • Charles Weibel, Survey of non-Desarguesian planes, Notices of the American Mathematical Society 54 (10): 1294–1303 (2007) pdf
category: algebra

Last revised on November 3, 2013 at 04:43:33. See the history of this page for a list of all contributions to it.