CHAPTER 1: INTRODUCTIO N 1.1. Geometric, Linear and Projective Transformations A geometric transformation g o f V ont o V x isabijectio n g:V+ V x whic h has the following property fo r all subsets X o f V: X i s a subspace of V i f and only if gX i s a sub- space of V x . It is clear that a composition of geometric transformations i s geometric, and that the in- verse of a geometric transformation i s also geometric. I f g: V * V x i s a geometric transforma - tion, the n g preserve s inclusion, join, meet, Jordan-Holder chains, among subspaces. S o we have the following proposition . 1.1.1. If g is a geometric transformation of V onto V x , then g(un W)=gun g w,g(u+ W)= g u+ g w, dimF gU = dimF U, £0=0, gV=V x holds for all subspaces U and W of V. By the projective space P(V) o f V w e mean the set of all subspaces of V. Thu s P(V) consists of the elements of po w V whic h are subspaces of V P{V) i s a partially ordered set, the order relation being provided by set inclusion in V an y two elements U an d W o f P(V) have a join and a meet, namely the subspaces U + W an d £/ n W, s o that P(V) i s a lattice P(V) ha s an absolutely largest element V, an d an absolutely smallest element 0 t o each ele- ment U o f P{V) w e attach the number dim F £/ eac h U i n P(V) ha s a Jordan-Holder chain 0 C • • • C U, an d all such Jordan-Holder chains have length 1 + dim F U. Define /»'(F)= {UeP(V)\dim F U=i} and call P 1 (F),i2(K),iy,-1(J0, th e set of lines, planes, hyperplanes, of V. A projectivity n o f V ont o V x isabijectio n TT:P(V) » P(V X ) whic h has the follow - ing property fo r all U, W in P(V): UCW i f and only if nUCnW. It is clear that a composition o f projectivities is a projectivity, and that the inverse of a projectivity i s also a projectivity. I f n:P(V)-P(V l ) i s a projectivity o f V ont o V l9 the n n preserve s order, join, meet, Jordan-Holder chains, among the elements of P(V) an d P(V X ). So we have the following proposition . 1.1.2. / / TT:P(V)+ P{V X ) is a projectivity of V onto V v then 2 http://dx.doi.org/10.1090/cbms/022/02

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