Quasifield

Quasifield

In mathematics, a quasifield is an algebraic structure (Q,+,.) where + and . are binary operations on Q, much like a division ring, but with some weaker conditions.

Definition

A quasifield (Q,+,.) is a structure, where + and . binary operations on Q, satisfying these axioms :

  • (Q,+) is a group
  • (Q_{0},.) is a loop, where Q_{0} = Q backslash {0}
  • a.(b+c)=a.b+a.c forall a,b,c in Q (left distributivity)
  • a.x=b.x+c has exactly one solution forall a,b,c in Q

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 (Q,+) is abelian.

Kernel

The kernel K of a quasifield Q is the set of all elements c such that :

  • a.(b.c)=(a.b).c forall a,bin Q
  • (a+b).c=(a.c)+(b.c) forall a,bin Q

Restricting the binary operations + and . to K, one can shown that (K,+,.) is a division ring .

One can now make a vector space of Q over K, with the following scalar multiplication : v otimes l = v . l forall vin Q,lin K

As the order of any finite division ring is a prime power, this means that the order of any finite quasifield is also a prime power.

Projective planes

Given a quasifield Q, we define a ternary map Tcolon Qtimes Qtimes Qto Q by

T(a,b,c)=a.b+c forall a,b,cin Q

One can then verify that (Q,T) satisfies the axioms of a planar ternary ring. Associated to (Q,T) is its corresponding projective plane. The projective planes constructed this way are characterized as follows: a projective plane is a translation plane with respect to the line at infinity if and only if its associated planar ternary ring is a quasifield.

References

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