代数结构与几何基础Ⅳ——仿射几何的公理体系  

Algebraic Structure and foundation of Geometry(Ⅳ)——The Axiom System of Affine Geometry

作　　者：梅向明[1] 侯忠义[1] 

机构地区：[1]首都师范大学数学系

出　　处：《首都师范大学学报（自然科学版）》1997年第2期1-5,共5页Journal of Capital Normal University：Natural Science Edition

摘　　要：在公理化方法定义的几何中引进“平行”关系,然后把结合公理I_8改成“平行公理”,我们就得到一种新的几何——仿射几何。本文将证明这种几何同构于某一体(域)上的n维仿射几何。若添加顺序公理,则这种几何同构于某一有序体(域)上的n维仿射几何,最后我们指出:三维仿射几何的结合公理、平行公理和顺序公理就是Hilbert公理体系中的结合公理、平行公理和顺序公理。The paper introduces the 'parallel 'relation in the geometry defined by the axiom system of projective geometry,then changes accidence axiom I8 for 'parallel axiom',and obtains a new geometry-affine geometry. The paper proves that this geometry is isomor- phic to the n-dimensional affine geometry on a field. If add the order axiom,this geometry is isomorphic to the n-dimensional affine geometry on an order field. Finally,the paper shows that the accidence, parallel and order axioms of the 3-dimensional affine geometry is just the same as the accidence,parallel and order axioms of Hilbert axiom system.

关 键 词：仿射几何 代数结构 同构 公理化方法 公理体系 证明 平行公理 顺序 基础 定义 

分 类 号：N03[自然科学总论—科学技术哲学] G633.63[文化科学—教育学]