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 $(A,+,\cdot)$ with two binary operations $+$ and $\cdot$ such that

• $(A,+)$ is a group with neutral element $0$
• $(A\backslash \{0\},\cdot)$ is a loop
• left distributivity $a\cdot(b+c) = a\cdot b + a\cdot c$ for all $a,b,c\in A$
• if $a,b,c\in A$, $a\neq b$ then the equation $a\cdot x = b \cdot x + c$ has a unique solution for $x\in A$

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