Ternary field

planar ternary ring

A set with two special elements, and , provided with a ternary operation satisfying:

A) for all ;

B) for all ;

C) if , , then there is a unique such that ;

D) if , then there is a unique such that ;

E) if , , then there are unique such that and .

Ternary fields were introduced in [a1] for the purpose of coordinatizing arbitrary, not necessarily Desarguesian, projective planes (cf. Desargues assumption; Desargues geometry; Projective plane). Slight variations of the original definition were given in [a2] and [a3], which is followed here. Given a projective plane, fix four points in general position: , , , , and let , and . For the points of one chooses coordinates with running over a set and , assigning to and to . The projection of from on is given coordinates , and then (see Fig.a1). The points on get one coordinate , with , or , where is an extra symbol , and the lines are coordinatized by , or , as in Fig.a2.

Figure: t092430a

Figure: t092430b

The ternary operation on is defined by if and only if lies on . The properties A)–E) for are then consequences of the axioms for a projective plane. Conversely, any ternary field coordinatizes a projective plane. It may happen that different ternary fields coordinatize the same plane, for a different choice of basis points , , , .

In case is finite, C) and D) are equivalent to D) and E); further, C) is then a consequence of D) and the existence of at most one as in C).

On a ternary field , addition is defined by ; with this operation is a loop with as neutral element. Multiplication is defined by ; this makes a loop with as neutral element. is said to be linear if for all , , . Linearity is equivalent to a very weak Desargues-type condition on triangles which are in perspective from the point (cf. Configuration, in particular Desarguesian configuration, and also Desargues assumption). Other algebraic properties of , such as associativity of addition or multiplication and left or right distributivity, can also be translated into certain Desargues-type conditions. In particular, a translation plane with as translation line, i.e., a plane in which the group of -translations is transitive on the points not on , is coordinatized by a (left) quasi-field, which is a linear ternary field with associative addition satisfying the left distributive law .

References

[a1]	M. Hall, "Projective planes" Trans. Amer. Math. Soc. , 54 (1943) pp. 229–277
[a2]	G. Pickert, "Projective Ebenen" , Springer (1975)
[a3]	D.R. Hughes, F.C. Piper, "Projective planes" , Springer (1973)