Semifields

We have discussed general coordinate systems for translation planes in chapter 12. We merely remind the reader of the relevant definitions of ‘presemifields’ and ‘semifields’.

Rem ark 25.2.1 A system (D ,+,o) is a ‘presemifield’ if and only if the

follomng axioms hold: (1 ) (D ,+) is an Abelian group; (2) The distributive laws are valid for x ,y ,z E D: (a) x o (y + z) = x o y + x o z \ (b) (y + z )o x = y o x - \-z o x . (3) (D*, o) is a quasigroup. A presemifield with identity is a isemifield) ((3/ (D*,o) is a loop).