### Related pages

Double loops
Coordinatization of affine planes
Coordinatization of projective planes
A pair (K,T) where K is a set and is a ternary operation on K is called a ternary field if the following four axioms hold.

• There are two distinct elements 0,1 in K with T(s,0,t)=t, T(s,1,0)=s, T(1,x,0)=x and T(0,x,t)=t for all s,x,t in K.
• Given x1, x2, y1, y2 in K with there exists a unique pair (s,t) in K2 with T(s,x1,t)=y1 and T(s,x2,t)=y2.
• Given four elements s1, s2, t1, t2 with , there exists a x in K with T(s1,x,t1)=T(s2,x,t2).
• For all s,x,y in K there exists a unique t in K with T(s,x,t)=y.
If (K,T) is a ternary field, we define addition and multiplication on K by the formulas

Then (K,+) and are loops, called the additive and multiplicative loop of (K,T), respectively.

The ternary field is linear if T(s,x,t)=sx+t for all s,x,t. For example, if K is a skew field, then T(s,x,t)=sx+t defines the structure of a linear ternary field on K.

Given any ternary field (K,T), we may construct an affine plane A(K,T) over (K,T). The points of A(K,T) are the points of the cartesian product K2 and the lines of A(K,T) are the vertical lines

and the lines of the form

All vertical lines are a single parallel class, and two lines [s1,t1] and [s2,t2] are parallel if and only if s1=s2.

The projective closure of A(K,T) is a projective plane, usually denoted by P(K,T). The new points of P(K,T) correspond to the parallel classes of A(K,T). The class of all lines with slope s is denoted by the symbol

and the class of vertical lines is

Finally,

is the line at infinity.