BOOLEAN FUNCTIONS: DIFFERENTIABLE MANIFOLDS AND AFFINE SPACES

The Boolean affine applications are the Boolean differentiable applications. If we accept to define the differentiable manifolds by replacing the differentiable structure with some atlas, then the affine spaces are special cases of differentiable manifolds. 0. Preliminaries (0.1) Definition},,1,0{2 ×Å=B is the field of the binary numbers, where ' ' Å is the modulo 2 sum and ' ' × is the intersection. It has the rough topology. (0.2) Definition The functions ': LLf ® , where ', LL are Boolean linear spaces, are called Boolean applications. From now on, we shall drop the attribute ’Boolean whenever possible. (0.3) Remark L becomes an affine space together with the function LyxxyxyyxyxLL def ÎÅ===' ´)()(), ( a where LLyx ®:, are translations. (0.4) Definition The affine applications f are these that satisfy: Lxxk k Î"Î " +121,...,,N,)(...)()... ( 121121 ++ ÅÅ=ÅÅ kk xfxfxxf (0.5) Definition If}| { Iiei Î is a base of L, then the first order derivatives of f are: Iixfexfxf ii ÎÅÅ= ¶),()()