In mathematics, a quasifield is an algebraic structure
, much like a division ring, but with some weaker conditions.
All division rings, and thus all fields, are quasifields.
, satisfying these axioms: Strictly speaking, this is the definition of a left quasifield.
A right quasifield is similarly defined, but satisfies right distributivity instead.
A quasifield satisfying both distributive laws is called a semifield, in the sense in which the term is used in projective geometry.
Although not assumed, one can prove that the axioms imply that the additive group
Thus, when referring to an abelian quasifield, one means that
The vector space construction implies that the order of any finite quasifield must also be a prime power.
All division rings, and thus all fields, are quasifields.
They are the finite fields of orders up to and including eight.
The smallest quasifields that are not division rings are the four non-abelian quasifields of order nine; they are presented in Hall (1959) and Weibel (2007).
satisfies the axioms of a planar ternary ring.
The projective planes constructed this way are characterized as follows; the details of this relationship are given in Hall (1959).
A projective plane is a translation plane with respect to the line at infinity if and only if any (or all) of its associated planar ternary rings are right quasifields.
It is called a shear plane if any (or all) of its ternary rings are left quasifields.
The plane does not uniquely determine the ring; all 4 nonabelian quasifields of order 9 are ternary rings for the unique non-Desarguesian translation plane of order 9.
These differ in the fundamental quadrilateral used to construct the plane (see Weibel 2007).
Quasifields were called "Veblen–Wedderburn systems" in the literature before 1975, since they were first studied in the 1907 paper (Veblen-Wedderburn 1907) by O. Veblen and J. Wedderburn.
Surveys of quasifields and their applications to projective planes may be found in Hall (1959) and Weibel (2007).