## Related pagesDouble loopsCoordinatization of affine planes Coordinatization of projective planes |

- 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 x
_{1}, x_{2}, y_{1}, y_{2}in K with there exists a unique pair (s,t) in K^{2}with T(s,x_{1},t)=y_{1}and T(s,x_{2},t)=y_{2}. - Given four elements s
_{1}, s_{2}, t_{1}, t_{2}with , there exists a x in K with T(s_{1},x,t_{1})=T(s_{2},x,t_{2}). - For all s,x,y in K there exists a unique t in K with T(s,x,t)=y.

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 K^{2}
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
[s_{1},t_{1}] and [s_{2},t_{2}] are
parallel if and only if s_{1}=s_{2}.

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.

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