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Lattice Theoretic Characterization of an Affine Geometry of Arbitrary Dimensions I

1952; Hiroshima University - Department of Mathematics; Volume: 16; Issue: none Linguagem: Inglês

10.32917/hmj/1557367259

0018-2079

Usa Sasaki,

Advanced Numerical Analysis Techniques

l (Vo1.16 P. 3. Every line contains at least three points.By a linear subspace of G, it is meant the subset S such that p, q ES imply p+qCS.Let S, T be subsets of G.If S, T=t=O, void set, we define S + T to be the set union V(p+q; p ES, q ET), where by p+p we mean the set !Pl containing p alone.If T=O we define S +T=T+S=S. 3> REMARK 1.1.It follows immediately from P. 2 that if p, q, r are points of G, then p+(q+r)=(p+q)+r, 4 > whence it may be denoted by p+q+r.DEFINITION 1. 2. Let A be a set of points.If for any pair of distinct points p, q, there exists a subset p vq (called line of A) containing p, q, and if for any triple p, q, r of non-collinear points, 5 > there exists a subset p v q v r (called plane of A) containing p, q, r, which satisfy the following conditions A.1-A. 4, then A is called an affine space. 6 'A.1.Two distinct points on a line determine the line.A. 2. Three non-collinear points on a plane determine the plane.Two lines p v q, r vs are called to be parallel to each other and denoted by p v q II r v s provided that they are contained in the same plane and have no point in common.A. 3.If p, q, r are non-collinear points, tMn there exists one and only one line r v s such that r v s is parallel to p v q.By a subspace of A, we mean the subset S such that p, q ES implies p V qCS, and p, q, r Es implies p V q V rCS.Let • T be any subset of A. The smallest subspace containing T is called the subspace generated by T.n If four points p, q, r, s are not on a plane, the subspace generated by the set of these points is called a 3-space and is denoted by p v q v r vs.A. 4. If two planes in a 3-space have a common point, then they have at least one more point in common.1952) 12) For the plane affine geometry, cf.Birkhoff [2] 110, Theorem 10.And for the finite dimensional affine geometry over a field, cf.ibid.103.13) Cf.Wilcox [1] 496.