Quasifield
In mathematics, a quasifield is an algebraic structure [math]\displaystyle{ (Q,+,\cdot) }[/math] where [math]\displaystyle{ + }[/math] and [math]\displaystyle{ \cdot }[/math] are binary operations on [math]\displaystyle{ Q }[/math], much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
Definition
A quasifield [math]\displaystyle{ (Q,+,\cdot) }[/math] is a structure, where [math]\displaystyle{ + }[/math] and [math]\displaystyle{ \cdot }[/math] are binary operations on [math]\displaystyle{ Q }[/math], satisfying these axioms:
- [math]\displaystyle{ (Q,+) }[/math] is a group
- [math]\displaystyle{ (Q_{0},\cdot) }[/math] is a loop, where [math]\displaystyle{ Q_{0} = Q \setminus \{0\} \, }[/math]
- [math]\displaystyle{ a \cdot (b+c)=a \cdot b+a \cdot c \quad\forall a,b,c \in Q }[/math] (left distributivity)
- [math]\displaystyle{ a \cdot x=b \cdot x+c }[/math] has exactly one solution for [math]\displaystyle{ x }[/math], [math]\displaystyle{ \forall a,b,c \in Q, a\neq b }[/math]
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 [math]\displaystyle{ (Q,+) }[/math] is abelian. Thus, when referring to an abelian quasifield, one means that [math]\displaystyle{ (Q_0, \cdot) }[/math] is abelian.
Kernel
The kernel [math]\displaystyle{ K }[/math] of a quasifield [math]\displaystyle{ Q }[/math] is the set of all elements [math]\displaystyle{ c }[/math] such that:
- [math]\displaystyle{ a \cdot(b \cdot c)=(a \cdot b) \cdot c\quad \forall a,b\in Q }[/math]
- [math]\displaystyle{ (a+b) \cdot c=(a \cdot c)+(b \cdot c)\quad \forall a,b\in Q }[/math]
Restricting the binary operations [math]\displaystyle{ + }[/math] and [math]\displaystyle{ \cdot }[/math] to [math]\displaystyle{ K }[/math], one can shown that [math]\displaystyle{ (K,+,\cdot) }[/math] is a division ring.
One can now make a vector space of [math]\displaystyle{ Q }[/math] over [math]\displaystyle{ K }[/math], with the following scalar multiplication : [math]\displaystyle{ v \otimes l = v \cdot l\quad \forall v\in Q,l\in K }[/math]
As a finite division ring is a finite field by Wedderburn's theorem, the order of the kernel of a finite quasifield is a prime power. The vector space construction implies that the order of any finite quasifield must also be a prime power.
Examples
All division rings, and thus all fields, are quasifields.
A (right) near-field that is a (right) quasifield is called a "planar near-field".
The smallest quasifields are abelian and unique. 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).
Projective planes
Given a quasifield [math]\displaystyle{ Q }[/math], we define a ternary map [math]\displaystyle{ T\colon Q\times Q\times Q\to Q }[/math] by
- [math]\displaystyle{ T(a,b,c)=a \cdot b+c \quad \forall a,b,c\in Q }[/math]
One can then verify that [math]\displaystyle{ (Q,T) }[/math] satisfies the axioms of a planar ternary ring. Associated to [math]\displaystyle{ (Q,T) }[/math] is its corresponding projective plane. 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).
History
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).
References
- Hall, Marshall Jr. (1959), Theory of Groups, Macmillan.
- Veblen, O.; Wedderburn, J.H.M. (1907), "Non-Desarguesian and non-Pascalian geometries", Transactions of the American Mathematical Society 8 (3): 379–388, doi:10.2307/1988781, https://zenodo.org/record/1431565
- Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS 54 (10): 1294–1303, https://www.ams.org/notices/200710/
See also
- Near-field
- Semifield
- Alternative division ring
- Hall systems (Hall planes)
- Moufang plane
External links
- Quasifields by Hauke Klein.
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