Affine Barbilian-Ebenen I

Abstract

Let P an abstract set of POINTS, G a subset of the powerset of P, whose elements we call LINES and Ø resp. ∥ two binary relations on P×P resp. G×G. An axiomatic characterization of those structures [P,G,Ø,∥] is given, which can be described as follows:

• 1. P={(x,y)¦ x, y ∃ R} R a Z-ring, i.e. a ring with identity 1 and the property a·b=1 iff b·a=1;

• 2. (x,y)Ø(x′,y′) iff (x−x′,y−y′) ∃ B, B a subset of p satisfying the conditions (E₁) to (E₄) below;

• 3. G={(a,b)+R(u,v)¦ (a,b) ∃ P, (u,v) ∃ B};

• 4. (a,b)+R(u,v) ∥ (c,d)+R(s,t) iff R(u,v)=R(s,t).

• (E1) (1,0), (0,1) ∃ B.

• (E2) r(u,v) ∃ B, whenever (u,v) ∃ B and r a unit in R.

• (E3) Each (u,v) ∃ B can be completed to an invertible 2×2-matrix ( ^{u v}_{s t} ) with (s,t) ∃ B.

• (E4) If (u,v), (s,t) ∃ B and ( ^{u v}_{s t} ) is invertible then (u,v)+ℓ(s,t) ∃ B for all ℓ ∃ r.

The class of these geometries covers besides the affine DESARGUES-PLANES for instance the affine ring-geometries considered by HJELMSLEV [4], [5], KLINGENBERG [7], [8], [9] and BENZ [2], [3].