Continuous Geometry 9781400883950

Table of contents :
Foreword
Table of Contents
PART I
Chapter I. Foundations and Elementary Properties
Chapter II. Independence
Chapter III. Perspectivity and Projectivity. Fundamental Properties
Chapter IV. Perspectivity by Decomposition
Chapter V. Distributivity. Equivalence of Perspectivity and Projec tivity
Chapter VI. Properties of the Equivalence Classes
Chapter VII. Dimensionality
PART II
Chapter I. Theory of Ideals and Coordinates in Projective Geometry
Chapter II. Theory of Regular Rings
Appendix 1
Appendix 2
Appendix 3
Chapter III. Order of a Lattice and of a Regular Ring
Chapter IV. Isomorphism Theorems
Chapter V. Projective Isomorphisms in a Complemented Modular Lattice
Chapter VI. Definition of L-Numbers; Multiplication
Appendix
Chapter VII. Addition of L-Numbers
Appendix
Chapter VIII. The Distributive Laws, Subtraction; and Proof that the L-Numbers form a Ring
Appendix
Chapter IX. Relations Between the Lattice and its Auxiliary Ring
Chapter X. Further Properties of the Auxiliary Ring of the Lattice
Chapter XI. Special Considerations. Statement of the Induction to be Proved
Chaptfr XII. Treatment of Case I
Chapter XIII. Preliminary Lemmas for the Treatment of Case II
Chapter XIV. Completion of Treatment of Case II. The Fundamental Theorem
Chapter XV. Perspectivities and Projectivities
Chapter XVI. Inner Automorphisms
Chapter XVII. Properties of Continuous Rings
Chapter XVIII. Rank-Rings and Characterization of Continuous Rings
PART III
Chapter I. Center of a Continuous Geometry
Appendix 1
Appendix 2
Chapter II. Transitivity of Perspectivity and Properties of Equivalence Classes
Chapter III. Minimal Elements
List of Changes from the 1935—37 Edition and comments on the text by Israel Halperin
Index

Citation preview

CONTINUOUS GEOMETRY

PR IN C E T O N L A N D M A R K S IN MATHEMATICS A N D P H Y S IC S

Non-standard Analysis, by A braham Robinson

General Theory of Relativity, by P.A.M. D irac

Angular Momentum in Quantum Mechanics, by A. R. Edm onds

Mathematical Foundations of Quantum Mechanics, by John von Neum ann

Introduction to Mathematical Logic, by Alonzo Church

Convex Analysis, by R. Tyrrell Rockafellar

Riemannian Geometry, by Luther P fahler Eisenhart

The Classical Groups, by Hermann Weyl

Topology from the Differentiable Viewpoint, by John W. M ilnor

Algebraic Theory of Numbers, by Hermann Weyl

Continuous Geometry, by John von Neum ann

Linear Programming and Extensions, by George B. D an tzig

Operator Techniques in Atomic Spectroscopy, by Brian R. Ju dd

CONTINUOUS GEOMETRY BY

John von Neumann

FOREWORD BY

ISRAEL HALPERIN

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

Copyright © 1960 by Princeton University Press Copyright renewed © 1988 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex L.C. Card: 59-11084 ISBN 0-691-05893-8 Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources First printing, in the Princeton Landmarks in Mathematics and Physics series, 1998 http://pup.princeton.edu Printed in the United States of America 1 3 5 7 9 10 8 6 4 2

FOREWORD

This book reproduces the notes of lectures on Continuous Geometry given by John von Neumann at Princeton. P art I was given during the academic year 1935—36, and Parts II and II I were given during the academic year 1936—37. The notes were prepared, while the lectures were in progress, by L. Roy Wilcox, and multigraphed copies were distributed by the Institute for Advanced Study. The supply was soon exhausted, and the notes have not been reproduced until now. In the present edition many slips in typing have been corrected, in P art I with the help of Wallace Givens. I have inserted a few editorial remarks and I have made a small number of changes in the text. These changes, together with some comments, are listed at the back of the book. Of these changes, only one is essential and it was authorized explicitly by von Neumann. Continuous geometry was invented by von Neumann in the fall of 1935. His previous work on rings of operators in Hilbert space, partly in collabo­ ration with F. J. Murray, had led to the discovery of a new mathem atical structure which possessed a dimension function (cf. F. J. Murray and J. von Neumann, On Rings of Operators, Annals of Mathematics, vol. 37 (1936), page 172, Case (H i)). The new structure had incidence properties resembling those of the system L n (L n denotes the lattice of all linear sub­ sets of an n —1 dimensional projective geometry), but its dimension func­ tion assumed as values all real numbers in the interval (0, 1). Von Neumann set out to formulate suitable axioms to characterise the new structure. It happened that just previously, K. Menger and G. Birkhoff had characterised L n by lattice-type axioms; in particular, Birkhoff had shown the structures L n could be characterised as the complemented modular irreducible lattices which satisfy a chain condition. Von Neu­ mann dropped the chain condition and replaced it by two of its weak consequences: (i) order completeness of the lattice (Axiom II on page 1 of this book), and (ii) continuity of the lattice operations (Axiom III on page 2 of this book). Lattices which are complemented, modular, irreducible, satisfy (i) and (ii), but do not satisfy a chain condition, were called by von Neumann: continuous geometries (reducible continuous geometries were considered later, in P art III). It is easy to see that in a continuous

geometry there can be no minimal element — th at is, no atomic element or point. The structure previously discovered by Murray and von Neumann in their research on rings of operators was an example, the first, of a con­ tinuous geometry. Von Neumann’s first fundamental result was the construction, for an arbitrary continuous geometry, of a dimension function with values rang­ ing over the interval (0, 1). The construction was based on the definition: x and y are to be called equidimensional if x and y are in perspective rela­ tion, th at is: for some w the lattice join and meet of x with w are identical with those of y with w. The essential difficulty is to prove th at the per­ spective relation is transitive. P art I of this book discusses the axioms for continuous geometry and gives the construction of the dimension function. These results were sum­ marized in von Neumann’s note, Continuous Geometry, Proceedings of the National Academy of Sciences, vol. 22 (1936), pages 92— 100. In a second note, Examples of Continuous Geometries, Proceedings of the National Academy of Sciences, vol. 22 (1936), pages 101—108, von Neumann used a simple device to construct an extensive class of new con­ tinuous geometries. He started with finite dimensional vector spaces over an arbitrary but fixed division ring 3 and showed how to imbed an ndimensional vector space in a ^-dim ensional one with k ^ 2 ; a countable number of repetitions of this imbedding procedure, followed by a metric completion, gives a continuous geometry. This construction uses a coun­ table number of factors k, each ^ 2, but the final result is independent of the particular choice of the k. This was shown by von Neumann in a manuscript (still unpublished): Independence of From The Sequence v (MS written in 1936—37). If 3 is the ring of all complex numbers, the continuous geometry obtained by the imbedding procedure is non-isomorphic to, and simpler than, the continuous geometry obtained from a ring of operators. This non-isomorphism is established in von Neumann’s paper The Non-Iso­ morphism Of Certain Continuous Rings, Annals of Mathematics, vol. 67 (1958), pages 485—496 (MS actually written in 1936—37). The imbedding procedure, the new examples, and the non-isomorphism theorem are not mentioned in the present book. Von Neumann’s next fundamental result was a deep and technically remarkable generalization of the classical Hilbert - Veblen and Young coordinatization theorem. This classical theorem asserts th at if n ^ 4, then the points of L n can be coordinatized with homogeneous coordinates, vi

the coordinates to be taken in some suitable division ring.®. Von Neumann first expressed this classical theorem in the following two equivalent forms which do not mention points explicitly and which coordinatize all the linear subsets in L n: L n, as a lattice, is isomorphic to (i) the lattice of all right ideals in ® n («®n denotes the ring of n-th order matrices with elements in Sf) and to (ii) the lattice of all right submodules of Q)n ( x*, which is non-singular in the sense th at x*x = 0 implies x = 0. The continuous geometries obtained from rings of operators do, in fact, possess such orthogonalizations; the Herm itian conjugation in the coordinatizing continuous ring coincides with the operation of taking the Herm itian adjoint operator in Hilbert space. In a continuation of P art II (not mentioned in this book) von Neumann studied continuous geometries which possess an orthogonalization. In particular, he analysed such geometries which, in addition, possess a tran ­ sition probability function P(x, a). This means: P(x, a) is defined for all x and all non-zero a in the geometry, 0 ^ P(x, a) ^ 1, and P(x, a) has certain properties characteristic of the transition probability function in quantum mechanics. Von Neumann showed th at every such geometry can be obtained from a ring of operators in a suitable Hilbert space (the dimensionality of the Hilbert space must not be restricted). This result is embodied in a m anuscript (still unpublished) Continuous Geometries With A Transition Probability (MS w ritten in 1936—137) and is im portant in applications to quantum mechanics. Von Neumann lectured on this topic at Princeton in 1937—38, but detailed lecture notes did not become avail­ able. In a continuation of P art II in another direction, von Neumann develop­ ed the theory of arithm etic of continuous rings. In a m anuscript (still un­ published {Arithmetics of Regular Rings Derived From Continuous Geome­ tries (MS written in 1936—37), von Neumann proved that every element in a continuous ring has a natural decomposition into a set of algebraic parts together with a purely transcendental part; he also gave other arith­ metic theorems. These results were summarized in a note Continuous Rings And Their Arithmetics, Proceedings of the National Academy of Sciences, vol. 23 (1937), pages 341—349. These results are not mentioned in the present book. P art II I of this book is concerned with lattices which are continuous geometries except that irreducibility is not assumed. The center of such a geometry may be any continuous Boolean algebra. Von Neumann analysed the reducible geometry relative to its center and introduced the basic concept of central envelope of an element a — th at is, the least element e in the center which satisfies e ^ a. He used central envelopes to establish the transitivity of perspectivity for reducible continuous geometries. Then viii

he began the construction of the dimension functions. At this point the lecture notes break off abruptly. In a letter dated November 12, 1936, von Neumann wrote: “ I can get the 'central decom­ position’ of the various dimension functions and in particular their exist­ ence, enumeration, etc., in ‘reducible’ continuous geometries, but I have not yet succeeded in decomposing these lattices themselves as completely as I could do it for bounded operator rings in Hilbert space. But I have some hope to do it.” It is not difficult to see how he m eant to get this central decomposition of the various dimension functions (cf. Page 294). But whether he succeeded in getting a satisfactory decomposition for the lattices themselves is not known to me. Von Neumann reviewed his previous work on continuous geometries in four colloquium lectures delivered before the American Mathematical Society, in September, 1937, at Pennsylvania State College, State College, Pa., U.S.A. Later, he began to write a systematic account of his research on continuous geometry which he planned to publish as a book in the American M athematical Society Colloquium Series. But his work in the theory of games, other interests, and the war, intervened. As the years went by, he finally decided th at the Princeton lecture notes, at least, should be reproduced. This is now accomplished with the publication of the present book. Kingston, Ontario, Canada December 14, 1959

I srael H alperin

Table of Contents F oreword by Israel H a lp e r in .................................................................

v

PART I Chapter Chapter Chapter Chapter Chapter Chapter Chapter

I. Foundations and Elementary Properties . . . II. In d e p e n d e n c e .............................................................. III. Perspectivity and Projectivity. Fundamental Properties...................................................................... IV. Perspectivity by D ecom position............................ V. Distributivity. Equivalence of Perspectivity and P r o je c tiv ity ................................................................. VI. Properties of the Equivalence Classes . . . . VII. D im ensionality..............................................................

1 8 16 24 32 42 54

P A R T II Chapter Chapter

Chapter Chapter Chapter Chapter Chapter Chapter

Chapter Chapter

I. Theory of Ideals and Coordinates in Projective G eo m etry ..................................................................... 63 II. Theory of Regular R in g s......................................... 69 Appendix 1 ................................................................. 82 Appendix 2 ................................................................. 84 90 Appendix 3 ................................................................. III. Order of a Lattice and of a Regular R in g . . 93 IV. Isomorphism T heorem s...................................................103 V. Projective Isomorphisms in a Complemented Modular L a t t ic e ..............................................................117 VI. Definition of L-Numbers; Multiplication . . . 130 A p p e n d ix ...........................................................................133 VII. Addition of L-Numbers...................................................136 A p p e n d ix ...........................................................................148 VIII. The Distributive Laws, Subtraction; and Proof that the L-Numbers form a R in g ........................... 151 A p p e n d ix ...........................................................................158 IX . Relations Between the Lattice and its Auxiliary R i n g ................................................................................... 160 X . Further Properties of the Auxiliary Ring of the ...........................................................................168 Lattice X

Chapter

XI.

Chaptfr Chapter

X II. X III.

Chapter

XIV.

Chapter

XV. XVI. XVII. X V III.

Chapter Chapter Chapter

Special Considerations. Statem ent of the Induc­ tion to be P r o v e d .................................................. Treatm ent of Case I .............................................. Preliminary Lemmas for the Treatm ent of Case I I ............................................................................... Completion of Treatm ent of Case II. The Funda­ mental T heorem ...................................................... Perspectivities and P ro jectiv itie s....................... Inner A u to m o rp h ism s.......................................... Properties of Continuous R in g s ......................... Rank-Rings and Characterization of Continuous R in g s ..........................................................................

177 191 197 199 209 217 222 231

PA RT III Chapter

I.

Chapter

II.

Chapter

III.

L ist

of

Center of a Continuous G eom etry..................... Appendix 1 .............................................................. Appendix 2 .............................................................. Transitivity of Perspectivity and Properties of Equivalence C la s s e s .............................................. Minimal E le m e n ts ..................................................

240 245 259 264 277

Changes from the 1935—37 Edition and comments on the

text by Israel H alperin.........................................

283

I n d e x ........................................................................................................................

297

xi

PART I

PART I • CHAPTER I

Foundations and Elementary Properties The basis of our discussion is a class L of elements a, b, c, •••, two or more in number, together with a binary relation < between pairs of elements of L . Unless otherwise specified, Axioms I—VI listed below will be assumed. A xiom I: Order. I x: a < a for no element a. I 2: a < b < c implies a < c. D e fin it io n 1.1: a > b means b < a\ a ^ b means a < b or a = b; a ^ b means b a. A xiom II. Continuity. I I X: For every set 5 7 , there is an element 27(5) in L, which is a least upper bound of 5, i.e. (a) 27(S) ^ a for every a in 5, (b) x ^ a for every a in 5 implies x ^ 27(S). I I 2: For every set S 272(5) and 272(5) > 27x(5), so I 2 implies 27x(5) > 27x(5), which is contradictory to l v In a similar manner, 77(5) is unique. D efin itio n 1.2: The elements 77(L), 27(7) will be denoted by 0, 1 respectively. It follows from H v (b) and I I 2,(&) th at if 0 is the em pty subset of 7 , 0 = 27(0), 1 = 7 7 (0 ). D efin itio n 1.3: Let (a, b) denote the class consisting of the elements a, b. Then we define a + b = 27(a, b),

ab = a • b = TI{af b).

Corollary : a + a = a, aa = a. D e fin it io n 1.4: Let Q be any infinite Cantor aleph. Let there be

W

2

PART I

•

CHAPTER I

given a system of elements (aa; a < Q). We consider only the two cases (i) a < P implies aa ^ a (ii) a < implies aa ^ afi. In case (i) we define hm* aa = Z (aa; a < Q), a-+Q

where the right member denotes the least upper bound of the set of all aa; a < Q; in case (ii) we define lim* aa = IJ(aa; a < Q). Corollary : Let (aa) be a system satisfying (i). Then for every b, the system (b + #«) satisfies (i) and

lim* (b

aa) = b + lim* aa.

a-+Q

Q

a—>Q

P roof : Consider the first part in which (i) holds. Clearly Z(b + aa; a < Q ) ^ b + aa ^ b and aa for all oc < 12 (use I I V (a)). Hence it is ^ Z ( a a; a < 12) and ^ b -\-E{a(t\ a Q

a—

I I I 2: Let 12 be an infinite aleph and(aa; a < 12) a system satisfying condition (i) of Definition 1.4. Then hm* (baa) = b hm* aa. a-^Si

a->Q

A xiom IV: Modularity.

IVX: a < c imphes (a + b)c = a + be for every b. A xiom V: Complementation. Vx: Corresponding to each element a of L there exists an element x in L such th at a

+

x =

1,

ax — 0.

FOUNDATIONS A N D E L E M E N T A R Y P R O P E R T I E S

3

The element x is referred to as an inverse of a. It is not assumed to be unique. A x io m VI: Irreducibility. VIX: If a has a unique inverse, then a is either 0 or 1. It is to be noted th at if in the axioms I—VI the relations < and > be interchanged, the statem ents thus obtained are equivalent to the original axioms, provided, of course, th at the accompanying substitution a)

>

Z

0

n

i

+

be made. This invariance of the postulates is referred to as the duality of our theory. Because of it, any theorem possesses a dual theorem ob­ tained from it by means of the substitution (1) and any other inter­ changes arising by subsequent definitions. The system L is seen to form a lattice in the terminology of G. Birkhoff.* Several further remarks concerning the axioms are in order at this point. First, it should be observed that Axioms I I X and I I 2 are not in­ dependent. In fact if I be assumed, then I I Xand I I 2 are equivalent. By duality, it suffices to show th at I I Ximplies I I 2. Let 5 be any subset of L , and let F be the set of all elements a x for every x in S. Then 27(F) ^ x for every x in S, and if a ^ x for every x in 5, then a e F, whence a 27(F). Hence I7(S) = 27(F) is effective in I I 2. Both IIj and I I 2 have been stated as axioms for purposes of symmetry. Secondly, it is worthy of note th at if a set L' is given which satisfies I but not II, then L' may be completed by a process** similar to the method of completing the rational numbers by Dedekind cuts, and the completion of L' will satisfy II. Thirdly, some explanation concerning the use of the term “irreducibility” in con­ nection with Axiom VI is necessary. This will be given at the end of the chapter. Finally, although this axiom is included at the outset of the theory, no use will be made of it until the latter part of Chapter VI. T heorem

(i) (ii) (iii) (iv)

1.1:

a -J~ b = b -j- a, ab = ba, (a -}- 6) -j- c = a -f- (b -f- c), (ab)c = a(bc). O ^a ^l. a + 0 = = a - l — a, a - 0 = 0, a + 1 = 1. 0^1.

* G. Birkhoff, Combinatorial relations in projective geometries, Annals of M athe­ m atics, vol. 36 (1935), pages 743— 748. ** H. M. M acNeille, Extensions of partially ordered sets, Proceedings of the National Academ y of Sciences, vol. 22 (1936), No. 1, pages 45— 50.

4

PART I

•

CHAPTER I

(v) a ^ b, b ^ a if and only if a = b. (vi) a+ ba = + a) = a. (vii) a + b = a if and only if a ^ b. (viii) ab = b if and only if a ^ b. (ix) If a 2^ b and c ^ d, then a + c ^ b + d and ac ^ bd. P roofs: Parts (i), (ii), (v) are immediate, and their proofs will be omitted. (iv) Suppose 0 = 1 . Then by (ii) 0 ^ a ^ 0 for every a. B y (v) a = 0 and L consists of but one element, contrary to the supposition that there are at least two elements. (vi) Clearly a + ba ^ a. But a ^ ba,whence a ^ a + ba. Hence by (v) a = a + ba. Dually, a(b + a) = a. (vii)Clearly a-\-b ^ b, hence a + b = a implies a ^ b . Conversely: a 2^ b implies by l l v (b), owing to a ^ a, that a ^ a + b. But a + b ^ a, so Ii gives a + b = a. (viii) This is the dual of (vii). (iii) This is immediate from (ii), (vii), (viii). (ix) a + c 2^ a ^ b, whence a + c ^ b. Similarly a + c ^ d. There­ fore a + c 2^ b + d.Dually, ac ^ bd. T heorem 1.2: If [a + c){b + d) — 0, then {a + b){c + d) = ac + bd. P r o o f: Define e = a -{- c, f = b + d. Then a ^ e, c rg e, b /, d ^ /, ef = 0. Now (a + f)(c + f) = (a + f)c + f (by IV) = (a + f)ec + / = (a + fe)c + / (by IV), and

(a + /) (c + /) = ac + /.

(2) Similarly,

(b -}- c) [d -{- e) = bd -f- c.

(3)

Also, by two applications of IV,

(a + f){b + e) = a + f(b + e) = a + b,

(4) and similarly,

(c + /) (d + e) = c + d.

(5) Therefore,

(a + b)(c + d) = (a + f)(b + e)(c + /)( p of J on K, not necessarily one-to-one. Suppose th at for each p € J,

(11)

x p ^ ap + ap, , ap. x p = 0.

Then (by 1).

Define oo C = 2 Xit i= 2

oo bn = 2 « ak3)pd f°r every perm utation i kt of the integers 1, 2, 3. L emma 4.1: Let a, b, c be given. Suppose oo

« = 2 «= J , b t ,

c = ^ ct ,

1= 1

1=1

1=1

where each sequence (a^ i = 1, 2, • • •), (&*•), (c*) is independent. If for each i = 1, 2, • • •, (ait bif c^pd, then (a, b, c)pd. P ro o f : By hypothesis, oo &i = ^ ;= 1

oo &ij *

=

^

oo ^ij >

;= 1

*

~ i= l

where for every i each of the sequences (atj; / = 1, 2, • • •), (&*•,), (c^) is independent. Hence by Theorem 2.4, the double sequences (a^; i, j = 1, 2, • • •), (bu), (cu ) are independent. Furthermore, ~ &i>~ ca ~ aa [24]

(h / =

2, • • •).

P E R S P E C T I V IT Y B Y DECOMPOSITION

25

Therefore, if we renumber the double sequences (ai3), (b{j)t (c^) into simple sequences (a'i )f (b*), (c^), respectively, we have a[ ~ 6' ~ c[ ~ a\

(i = 1, 2, • • •),

K-; i = l , V - - ) _ L , ( K) ±, (c'i)±. oo

oo

oo

t = Z K ’ i=i

* = 2 *i', i=i

c = 2 c'i> *=i

and (a, 6, c)pd. C o r o lla r y : If (ax, 6X, c^)pd, (a2, b2, c2)pd, and = &x&2 = cxc2 = 0, and a = + a2, & — bx + b2, c = cx + c2, then (a, b, c)pd. P r o o f : Define a{ = b{ = ct- = 0 (i = 3, 4, • • •). Then Lemma 4.1 ap­ plies and (a, 6, c)pd. L emma 4.2: Let two sequences (a^ i = 1, 2, • • •), (a'; * = 1, 2, • • •) be such th at (1)

= 0,

« 2)> l j = 1, 2, • • •) JL for i = 0, 1, 2, and

(15)

aoj~ ai

^oj

(/ = 1, 2, • • •)•

Thus if w = 1, a0 and am+1 are expressed as the sums of independent sequences, corresponding terms of the sequences being perspective. Suppose m > 1 and let P { be a perspective isomorphism of L (0, a{) and L(0, ai+1) (i — 2, • • •, m). The independent elements a2j ^ a2 are thus mapped by P 2 into independent elements azj ^ az, which in turn are mapped by P z into independent elements azj 5^ az, etc.; finally, the ami ^ am are mapped by P m into am+M ^ aw+1. Hence (a*,; / = 1, 2, • • •) _L (i = 3, • • •, w + 1), and we have oo

= 2

(* = 3, • • •, w + 1).

1=1

Moreover, azj~ • • • ~

(16) But by (15),

(/ = 1, 2, • • •)•

an(i we may compress (16) into a2, ~

(!7)

~ *«+m

(/ = 1, 2, • • •);

obviously am+i,j ^ aoam+i — Hence the induction hypothesis applies to the sequences of length m in (17), and we may conclude aoj~ am+l j . Therefore, in the case m > 1 also, a0 and am+1 are expressed as sums of independent sequences with corresponding pairs of terms perspective. Since oo

oo

2 a0i 2 am+l,i “ aQam+l — 1=1

1=1

the combined system (a01, a02, • • •, am+l lf am+12, • • •) is independent, and a0~ am+1 by Theorem 3.6. This completes the induction. Theorem 4.2: If ab = 0, then a & b if and only if b. P ro o f: It has already been noted (Corollary 2, Definition 3.5) that a ~ b implies a ?&b. Suppose a & b. Then there is a sequence (a0, • • •, an) with a0 = a, an = b, a, _ i ~ a* (i = 1, • • •, n). Since a0an = 0, we have by Lemma 4.7, a0~ a n, i.e., a ~ b .

Theorem 4.3: Let (a*; i — 1, 2, • • •) be an infinite independent sequence. If for every i — 1, 2, • • •, a{ ai+lt then each = 0.

P E R SP E C T IV IT Y B Y DECOMPOSITION

P roof : Clearly the pair (ait ai+1) is independent, whence atai+1 = 0 (i = 1, 2, • • •)• Therefore by Theorem 4.2, ai+1 for i = 1, 2, • • *, and by Theorem 3.8 each = 0. T heorem

4.4: If a & b, a ^ b, then a = b.

P roof : Define b[ = a ^ b , and let bx be an inverse of b[ in b. If P

is a projective isomorphism of L (0, a) and L (0, b), we obtain images av ax in L (0, a) of ^ under P . If a

and let ai+1, aM be the images of bi+1, b'i+1 under P in L (0, b). Thus there are defined four infinite sequences (a*; i = 1, 2, • • •), (bt), (a'.; i = 0, 1, • • •), (b't), which have the following properties: (b) (c)

at is inverse to a[ in a\_x (a'0 = a), b{ is inverse to b\ in b (6' = b) & bit a[ & b[

(i = 1, 2, • • •), (i = 1, 2, • • •).

From (b) we conclude immediately = 0,

bM £ b ' i9

b'M ^ b '

(i = 1, 2, • • •).

B ut these conditions imply (b i = 1, 2, • • •) J_ by Lemma 4.2. Now by (c) and (a) we have b{ & a{ = bi+1 (i = 1, 2, •• •). Hence by Theorem 4.3, b{ = 0 (i = 1, 2, • • •). In particular, bx = 0. Since bt is an inverse of a in 6, a = b + bx = b.

P A R T I • CHAPTER V

Distributivity, Equivalence of Perspectivity and Projectivity D e fin it io n 5.1:

(a, bt c)D means

a)

(a + b)c — ac + bc\

(1)

(a, b)D means {a, b, c)D for every c in L, i.e., (1) holds for every choice of c; c) (a)D means (a, b)D for every b, i.e., (1) holds for every choice of b, c. C o r o lla r y : {a, 6, c)D is eq u ivalen t to (b, a, c)D; (a, 6)Z> is equivalent to (b, a)D. b)

T h e o r e m 5.1: The relations D defined in Definition 5.1 are self-dual

and symmetric. P r o o f : It suffices to prove the theorem for the ternary relation D defined in part a), since it then automatically follows for the binary and unary relations of b) and c). Let D' be the dual relation to the ternary relation D , so that (a, b, c)D' in case ab + c = (a + c)(b + c). It will first be shown that (a, b, c)D implies (b, c, a)D’. Suppose (a, b, c)D, i.e., that (1) holds. Then (b + a)(c + a) = a + (a + b)c

(by IV)

= a + ac + be = a -(- be, whence (&, c, a )D \ Dually, (a, b, c)D' implies (b, c, a)D. Hence from a statement of the form (a, b, c)D or (a, b, c)D' we may infer one obtained by simultaneously interchanging D and U and performing the cyclic permutation of the letters which replaces each letter by its right neighbor. Hence, by three applications of this principle, (2)

(a, bt c)D

(b, cf a)D‘ -> (c, a, b)D -> (a, b, c)D'.

From (2) we may conclude that D is self-dual, since (a, b, c)D implies (a, b, c)D', and hence by duality (a, b, c)D' implies (a, b, c)D. We have V2]

P E R S P E C T I V IT Y AND PROJECTIVITY: EQUIVALENCE

33

also from (2) th at (a, b, c)D implies (c, a, b)D; this states th at a statem ent of the form (a, b, c)D implies the statem ent obtained from it by performing the cyclic perm utation on the letters which replaces each letter by its left neighbor. By this result and the Corollary to Definition 5.1, we have {a, b, c)D ^(c, a, b)D-+(b, c, a)D ^(c, b, a)D->(a, c, b)D->(b, a, c)D ^(a, b, c)D, which proves the sym m etry of D. Tacit use of Theorem 5.1 will be made constantly in the sequel, the frequency of its application prohibiting references to it. In the discussion of Axiom VI in Chapter I, it was seen (Theorem 1.5) th at this axiom is equivalent to a property which resembles algebraic irreducibility. Since this notion and the accompanying notion of direct sum are vital in what follows, we shall give at this point precise definitions of them. D e f i n i t i o n 5.2: The lattice L is said to be the direct sum of L (0, a) and L (0, b) (L — L (0, a) © L (0, b)) in case ab = 0 and each element x of L is expressible in the form x = u + v, with u ^ a, v ^ b. We shall say that L is reducible if there exist elements a, b, both distinct from 0, 1, such th a t L = L (0, a) © L ( 0, b), and irreducible if it is not reducible. Ir­ reducibility is the property denoted in Chapter I by J . C o r o l l a r y : If L = L (0, a) © L ( 0, b), then (a)

a

b — 1, and

(b)

each element x is expressible uniquely in the form x — u + v, u ^ a , v ^ b; in fact u — ax, v — bx.

(a) Since 1 is of the form u + v, we have 1 ~ u + v ^ a - \ - b , whence a + b = 1. (b) See Lemma 1.2. P roof:

T h e o r e m 5.2: L — L (0, a) ©Z,(0, b) if and only if a is inverse to b and (a, b)D. P r o o f : By Definition 5.2, L = L (0, a) © L (0, b) implies ab = 0, a + b = 1 and x = u + v = ax + bx for every x. Hence x = 1 • x = — (a + b)x = ax + bx and (a, b)D. Conversely, let ab = 0, a + b = 1, {a + b)x — ax + bx for each x. Then x —ax + bx — u + v, with u = ax ^ a, v = bx fg b. T heorem

5.3: The following statem ents are equivalent to each other:

(a)

a has an inverse b, for which (a, b)D.

(b)

a has a unique inverse.

(c)

(a)D.

PART I

34

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CHAPTER V

We prove th at {a) (6), and th at (a) ^ (c). (а) -> (6): See Theorem 5.2 and Lemma 1.2. (б) -> (a): See Theorems 1.5 and 5.2. (a) (c): By Theorem 5.2, L = L (0, a) © L (0,6). Hence for every x, y, x — u + v, y = ux + vv with u, ux fg a and v, vx b. Consequently x + y = (u + Ui)-\~(v + ^i)> where u + ux ^ a and v + vx ^ b.Thus by Lemma 1.2 (or by the Corollary to Definition 5.2), u = ax, ux = ay, u + ux — a(x + y), and therefore a(x -f y) = ax + ay. This proves (a)D. (c) -> (a): Let b be an inverse of a. Since (a)D, therefore (a, b)D. P roof:

T h e o r e m 5.4: If (a)D, (b)D, and if a' is the unique inverse of a, then (a + b)D, (ab)D and (a')D. P r o o f : ab(x y) = a(bx - f by) = abx + &by, whence (ab)D. By duality (a + b)D. The inverse a' of a is unique by Theorem 5.3, (b), (c), and both (a)D and ( axx = bxx — 0. Hence x

ax + bx, and (ax, bx)D by Theorem 5.6. Thus x = x(ax + 6X) = xax + xbx = 0 + 0 = 0.

Then ax =

bx ^ ab = 0, so ax — bx — 0.

Corollary : (a, b)D, ab = 0 if and only if

(5)

ax ^ a, bx

b, ax

bx implies ax = bx = 0.

P roof : The forward implication is evident since if ab = 0, then also

PART I

36

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CHAPTER V

a1b1 = 0, and ax bx implies ax ~ bx by Theorem 4.2, whence Theorem 5.7 applies, yielding ax = bx = 0. Conversely (5) imphes (4) and hence implies (a, b)D, ab = 0, by Theorem 5.7. We are now in a position to clarify to some extent the motivation of the present discussion. Since this theory is similar to the theory of sets, one would expect in it an analogue of the theory of comparability. It is natural in the light of Theorem 4.4, which states that a ^ b & a implies a = b, to define a < b as the existence of b' such that a & b' < b. (The corresponding relation in set theory is “a has smaller power than b”.) One expects the relations a < b , a ^ b, &< a to be exhaustive. This matter is, however, closely connected with the question of irreducibility, as is seen by the following consideration. Suppose that L is reducible, i.e., is a direct sum L (0, a) ©Z,(0, b), a =£ 0, b ^ 0. Then by Theorem 5.2, (a, b)D, and ax ^ a, bx ^ b, ax bx imphes ax = bx = 0 by the Corollary to Theorem 5.7. If now a < b, there exists bx ^ b with a & bv and a = bx = 0, which is impossible. Similarly neither a b nor b < a holds. This shows that Axiom VI must hold if comparability holds. The converse to this, viz., that VI imphes comparabihty, is one of the essential results to be obtained at the end of the present chapter. L e m m a 5.1: If {av b)D, (a2, b)D, axb = a2b = 0, then (ax + a2, b)D and (ax + a2)b = 0. P r o o f:

For every x (ax + a2 + x)b = axb + (a2 + x)b

(since (av b)D)

= axb + a2b + xb

(since (a2, b)D)

= K + &2 )b + xb

(since {a2> b)D).

Hence (ax + a2, b)D. Clearly (ax + a2)b = axb + a2b = 0. L e m m a 5.2: Let Q be any ordinal and (aa; a < Q) a system such that a < ft imphes aa ^ a If

(aa> b)D, aab = 0

(a < Q )f

then (hm* aa, b)D, (hm* aa)b = 0. a —►O

oc—yQ

P ro o f:

(lim* aa)b = lim* baa a—+Q a-~+Q = hm* 0 = 0. a-+Q

(by III2)

P E R S P E C T IV IT Y AND PROJECTIVITY: EQUIVALENCE

37

Also, for every x (b + lim* aa)x = (lim* (b + aa))x a—*G a.-+Q = lim* (b + aa)x

(by I I I 2)

= lim* (bx + aax) C L —

= bx + lim* aax = bx + (lim* aa)x a—

(by III2)

c l -+£)

and the proof is complete. T heorem 5.8: If (a, b)D, ab = 0, there exists a* ^ a such that (a*, b)D, a*b = 0 and a* is maximal with this property, i.e., if a' ^ a*, (a', b)D, a'b = 0, then a' = a*. P r o o f : For the purposes of this proof we shall write (x, y )P in case (x, y)D and xy — 0. Let Q be the first ordinal corresponding to the first power greater than the power of L. Define afi for every ordinal /? < Q by induction: define a0 = a and if aa has been defined with (aa, b)P for all a < /S, proceed as follows: I. Let /? have a predecessor a 0 so th at /? = a 0 + L If aaQ is maximal such th at (aa^ b)P, define = aa^. In the contrary case, there exists Up > aaQ such th at (afi, b)P. II. Let {$ be a limit ordinal. Then define afi = E(aa\ a < /?), whence (afi, b)P by Lemma 5.2. If a < ( } < Q and aa is not maximal such th at (aa , b)P, then it is evident th at aa < ap. If for every a < D, aa fails to be maximal such th at (aa, b)P, then the aa are all distinct, and their class has power greater than the power of L, which is impossible. Hence there exists an ordinal a x for which aai is maximal such th at (~>a± ^ a' 5^ a, th at is a, bx ^ 6, a1 ^ Therefore the Corollary to Theorem 5.7 gives ax = b1 = 0. Now cx~ ax == 0, cx = 0. Thus we have = bx = 0 under these assumptions. Therefore Theorem 5.7 yields (c, b)D, cb = 0. T heorem 5.9: If (a, b)D, ab = 0, and if a* ^ a is the maximal element

such th at (a*, b)Dt a*b = 0 (cf. Theorem 5.8 and its Corollary 2), then (a*)D. P roof : Let b* be inverse to a* and consider ax a*, bx ^ 6*, ax~ We have (a*, b)D, a * b = 0 and b1^ ax ^ a*; therefore Lemma 5.3 gives (bv b)D, bxb = 0. Now Corollary 1 to Theorem 5.8 implies bx ^ a*. But at the same time bx b*, so th at bx fg a*b* = 0, a1~ b 1 = 0 ; th a t is ax = = 0. Therefore Theorem 5.7 yields (a*, b*)D, and by Theorem 5.3, (a), (c), we have (a*)D. At this point we assume Axiom VI, the irreducibility of L. Unless otherwise specified, all six axioms will be assumed henceforth. T heorem 5.10: If (a, b)D, ab = 0, then a = 0 or b = 0. P ro o f : Let a* ^ a be maximal with the property (a*, b)D, a*b = 0. Hence (a*)D by Theorem 5.9, i.e., a* has a unique inverse by Theorem 5.3, (a), (b). Axiom VI then yields th at a* = 0 or a* = 1. If a* = 0, then a = 0 ; if a* = 1, then

b = 1 • b = a*b = 0. T heorem 5.11: (a, b)D if and only if a ^ b or b a. P roof : Suppose a ^ b or b ^ a. I f a ^ 6, a b = b and ax ^ bx,

ax + bx = bx. Therefore (a + b)x = bx = ax + bx. Thus (a, b)D. The case b ^ a is similarly treated. Conversely, suppose {a, b)D. Let a1 be an inverse of ab in a, and bx an inverse of ab in b; thus ax ^ a, bx b, and ax • ab = bx • ab = 0. B ut (av bx)D, axbx = 0, by Theo­ rem 5.6, whence ax = 0 or bx = 0 by Theorem 5.10. Hence either a = ab + ax = ab

b,

b = ab + bx = ab

a,

or and the proof is complete.

P ER SP E C TIV IT Y AND PROJECTIVITY: EQUIVALENCE

39

T heorem 5.12: If L satisfies Axioms I—VI, and if a ^ b, then L(a, b)

also satisfies these axioms. P roof : It has already been shown (Corollary to Theorem 1.3) th at Axioms I—V hold for L(a, b) if they hold for L. It remains to prove th at if L is irreducible then L(a, b) is also irreducible. I. a — 0. It will be shown that if 1,(0, b) = L (0, c) ® L ( 0, d), then c — 0 or d = 0. Now cd = 0 by the definition of direct sum; by the Corollary to Definition 5.2, c + d = b and x ^ b implies x = cx + dx. Let y be any element in L. Then y{c + d) -ijLc + d, and y(c + d) = y(c -f d)(c + d) — y(c + d)c + y(c + d)d = yc + yd, whence (c, d)D. Hence c = 0 or d = 0 by Theorem 5.10. II. 6 = 1. The lattice L(a, 1) is irreducible by duality. III. Consider now the general case. Since L(0, b) is irreducible, it satisfies all the axioms, and hence property of L deducible from these axioms is possessed also by L (0, b). In particular, the property proved in II above is possessed by L(0, b). But this is precisely the statem ent th at L(a, b) is irreducible, since 1 must be replaced by b when passing from L to L (0, b). T heorem 5.13: If ab = 0, a 0, 6 ^ 0 , then there exist a', b' such that 0 ^ a' ^ 0 =£b' iJLb, and b'. P roof : If (a, b)D, then a = 0 or b = 0 by Theorem 5.10, contrary to

the hypothesis. Hence (a, b)D is false. Theorem 5.5 then applies, yielding the desired result. T heorem 5.14: Let ab — 0. Then either there exists b' such th at a ^ b' ^ b, or there exists a' such th at b ~ a' ^ a. P r o o f: Let Q be the first ordinal corresponding to the first power greater than th at of L. We shall proceed to construct two independent transfinite systems [aa\ a < a 0), (ba; a < a0), such th at aa ^ a, ba ^ b, ba (a < a 0 < Q), and such th at either 2 (a a; a < a0) = a or 27(6a; a < a 0) = b. Suppose (afi; fi < a) J_, (by, fi < a)_L have been defined, and th at a^ ^ a, bp f ^b, a^r ^bp (ft < cl). It is then desired to define aa, if possible. If either S(ap; /? < a) = a or £{bp] ft < c l ) = b, then neither aa nor ba is to be defined. In the contrary case when neither of these conditions holds, there exist inverses ca and da of H>(ayf fi < a) and £(bp) /? < a) in a and b respectively, such that ca 0, da ^ 0. Then cada ^ ab = 0, and there exist aa, ba such th at 0 ^ aa ^ ca> 0 ^ ba ^ da and aa~ ba by Theorem 5.13. If a' is such th at all the a* (and hence

PART I

40

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CHAPTER V

all the ba) are defined for a < b.

Corollary : The relations < , >

are dual to each other. P r oof : This is evident, since the relation ~ is self-dual and the rela­ tions > , < are dual to each other. T heorem 6.1:

(a) (b) (c) (d)

a > b if and only if b < a. if a a! , b b', then a < b isequivalent to 6,

P roof :

(a) Let a > b. Then there exists b* such th at b* > b. Let a* be a perspective image of b in L(0, a). Then b ~ a* < a, and b < a. The converse is dual. (b) It suffices to show th at a < b implies a' < b'. If a •< b, there exists b* such th at b* < b. Let b* be a perspective image of b* in L ( 0, b'). Then a ' ~ b * ~ b* < b', whence # '< & ', since the relation ~ is transitive. (c) By hypothesis there exist b*f c* with b* < b >—' c* < c. Since b ~ c * , b* < b, there is a perspective image of b*, c** < c, and a < c. (d) By Theorem 5.15 and (a) of the present theorem one of the three relations a > b, b, a < b m ust hold. If two of them hold simultaneously we have b < a, b, or a < b, b, or a < 6, b < a. Using (a), (b), [42]

P R O P E R T I E S OF T H E EQ U IV AL EN CE CLASS ES

43

(c), we obtain a < a in each case; th at is, the existence of a* such th at a ~ a* < a, which contradicts Theorem 3.7. Let us consider a lattice L(a, b). Since it satisfies the same axioms as L satisfies, the relations of perspectivity and projectivity may be defined in L(a, b) in the same way as in L. Now let c, d e L ( a , b). We propose to investigate the question whether or not L(a, b)-perspectivity of c and d is equivalent to L-perspectivity of c and d. The latter means by Theorem 3.1 the existence of x e L such th at c + x = d + x = b, cx = dx — a. B ut this means the existence of a common inverse in L(a, b) of c, d, i.e., L{a, b)-perspectivity. Hence perspectivity in L and perspectivity in L(a, b) coincide. For projectivity this is not so obvious. However, since perspectivity and projectivity are equivalent (in L(a, b) as well as in L), our result extends also to projectivity. D e fin it io n 6.2: Let A a denote the class of all elements x such that x ~ a , and let denote the class of all A a, a e L . Corollary : The system (Aa; a e L ) is a m utually exclusive and exhaus­ tive partition of L into subclasses, i.e. (a) (b) (c)

A a = A b if and only if a r^j b\ A a ^ A„ implies $ ( A a, A b) = &; a e A a for every a e L .

P roof : This follows from the fact th at the relation ~ symmetric and transitive. The elements of will be denoted by A, B, C, • • *.

is reflexive,

D efin itio n 6.3:

(a) (b)

A < B means th at there exist a e A, b e B A > B means th at there exist a e A, b e B

with a < 6. with a > b.

C o r o ll a r y 1: A < B is equivalent to B > A ; the relations > , < are dual to each other. P roof : This follows from Theorem 6.1(a) and the Corollary to Defini­

tion 6.1. Corollary 2: A < B if and only if a < b for every a, b such that a e A, b e B. P roof : This is obvious by Theorem 6.1 (b). C o r o lla r y 3: A < B if and only if for each a e A there exists b e B such that a < b. P roof : Suppose A < B, and let bx be any element of B. Then a < bv by Corollary 2, and there exists b with b1~ b > a, whence b e B.

44

PART I

•

C H A P T E R VI

Conversely, let a be any element of A and b the corresponding element of B with a < b. Then a < b and A < B by definition. Corollary 4: A < B if and on ly if there exist a e A, b e B w ith a < b. P ro o f : This is obvious from Corollary 3. Corollary 5: For every A, B one and only one of the relations A > B, A — B t A < B holds. P roof : This is evident by Theorem 6.1 (d), considering Definition 6.2, Corollary (a), and Corollary 2 above. Corollary 6: is linearly ordered by the relations < , > . P roof : The relation < is transitive by Theorem 6.1 (c) and Corollary 2 above; > is transitive by Corollary 1. The linear ordering follows by Corollaries 1, 5. L emma 6.1: If a ~ b and a', b' are inverses of a, b, respectively, then a' ~ b'. P ro o f : Since a ~ b , there exists x with a + 2c = ft + « = l , ax = bx — 0. B ut a' + a = 1, a'a — 0, whencex. Similarly, bf ~ x, and a!~ b' by the transitivity of L emma 6.2: If a c, b ^ c, ~ b, and if a', br are inverses in c of a, b, respectively, then a' ~ b'. P roof : This follows from Lemma 6.1 b y replacing L b y L(0, c). L emma 6.3: If a ^ c, b d, a ~ b , d, and if a’, b' are inverses of a, b in c, d, respectively, then a’^ b ' . P roof : Let bv b[ be perspective images in L(0, c) of b, b', so th a t 6 ~ bx c, bf ~ b [ ^ c and b[ is inverse to b1 in c. Hence b1 and Lemma 6.2 applies, yielding B ut whence a ' ~ b ' . L emma 6.4: If ab = 0 and c is an inverse of a + b, then a is an inverse of b + c, and b is an inverse of c in b + c. P roof : Since ab = 0, (a + b)c = 0, it follows th at (a, b, c) J_. Hence a(b + c) = 0, and by hypothesis a + b + c = 1; therefore a is an in­ verse of b + c. Also since be = 0, b is inverse to c in b + c. L emma 6.5: If ab = 0, ef = 0, a ~ e, b ~ /, then a + b ~ e + /. P roof : Let c, g be inverses of a + b, e + / respectively. Then b y Lemma 6.4, b + c is inverse to a, and / + g is inverse to e. Thus b + /+ g by Lemma 6.1. Also by Lemma 6.4, b is inverse to c in b + c> and / is inverse to g in / + g. Therefore g by Lemma 6.3, and a -f* e+ / by Lemma 6.1. D e fin it io n 6.4: We shall say th at A + B exists in case there exist a e A, b e B such th at ab = 0. When A + B exists it is defined as the

P R O P E R T I E S OF T H E E Q U IV A L E N C E CL ASS ES

45

unique class C which is equal to A a+h for every a, b such th at a e A , b e B, ab = 0 (the existence and uniqueness of C following from Lemma 6.5). Thus A + B depends on A and B only. We shall say that A — B exists in case there exist a € A , b e B such th at a ^ b, i.e., in case A ^ B. When existent, A — B is defined as the unique class C which is equal to A b,, for every b' such th at there exist a e A, b e B, b ^ a, for which b' is inverse to b in a (the existence and uniqueness of C following from Lemma 6.3). Thus A — B depends only on A and B. We agree th at the assertion of any property of A + B means first the assertion of the existence of A + B, and secondly th at the class A + B as defined in Definition 6.4 has the stated property. In' m any cases where the first part is trivial its proof will be omitted. The same convention will be adopted for A — B. D e fin it io n 6.5: We define 6 = A 0, f = A v Corollary 1: Q ^ A ^ f for every A in . Corollary 2: f — A exists for every A. T heorem 6.2:

(a) (b)

A — B exists if and only if A ^ B, A + B exists if and only if f — A ^ B.

P ro o f : P art (a) is obvious. To prove (b), suppose first th at A + B exists. Then there exist a, b with a € A, b e B, ab = 0. By the Corollary to Theorem 1.4, there is an inverse bx of a such th at bx ^ b. Hence bx € (f — A), and f — A ^ B. Conversely, let f — A ^ B. Ii a is any element of A and if b± is an inverse of a, then there exists b € B such th at bx ^ b. Hence ab abx — 0. T heorem 6.3:

If A + B exists, then A + B = B + A. If A + B, {A + B) + C exist, then (A + B) + C = A + ( £ + C). A + 6 = A. If A — B exists, then (A — B) + B = A. If A + B exists, then (A + B) — B = A. If A + B exists, then A ^ A + B. The equation A + X = B has a solution if and only if A B, in which case the solution is unique. (viii) A < B is equivalent to the existence of X ^ 6 such th at B — A + X. (ix) If A + C, B + C exist, then A = B are equivalent to A + C = B-\-C respectively.

(i) (ii) (iii) (iv) (v) (vi) (vii)

PART I

46

•

C H A P T E R VI

(x)

If A 5^ C, B D, and if C + D exists, then A + B exists and A + B ^ C + D.

(xi)

A ^ B are equivalent to } — A = \ — B respectively.

P roof :

(i) This is trivial. (ii) Since A + B exists, there are elements a e A, b e B with ab — 0, and since (A + B) + C exists, there are elements ce C, d e (A + B) with cd = 0, d ~ a + b. Let av bx be perspective images of a, b in L(0, d). Then ax e A ybx € B, axbx — 0; also d — ax + bv Now bxc 5^ cd = 0, and B + C exists; also ax(bx+c) = 0 by Theorem 1.2, w hence^ + (B + C ) exists. Finally, A + (B + C) = 4 0t+(»1+c, = A {tti+bi)+e = (4 + B) + C. (iii) Clearly A + d exists, since a- 0 = 0 for every a e A . Hence A + 6 = A a+0 = A a = A. (iv) Since A — B exists, there are elements a e A, b e B with a ^ b . Let bx be an inverse of b in a. Then bbx = 0, b + bx = a, and A — A a — A h+h^ = A b + A b^ = B + (A — B). (v) Let ab = 0, a € A, b e B. Then A + B = A a+b, (A + B) - B = A a+b — B = A a = A, since b is inverse to a in a + b. (vi) This is obvious, since always a 5^ a + b. (vii) Suppose A + X == B has a solution. Then by (vi), A t S ^ A+ X == B . On the other hand, if A ^ B, then by Theorem 6.2(a), B — A exists, and X = B — A is effective in A + X —B by (iv). Suppose now th at X is a solution. Then B — A — (A + X ) — A = X by (v), whence the solution is unique. (viii) A ^ B is equivalent to the existence of X with B = A + X by (vii). Also, by (vii) (the uniqueness of X), A — B if and only if X = 0. Hence A < B if and only if X ^ 0. (ix) By (viii), A < B is equivalent to the existence of X ^ 0 with B = A + X. Hence B + C = ( A + X ) + C = ( A + C ) + X (by (i), (ii)), i.e., A - \ - C < B + C b y (viii). Interchanging A, B, we see th at A > B implies A + C > B + C. Finally, A = B obviously implies A + C = B + C. Hence A = B imply A + C = B + C respectively. Since we have exhaustive disjunctions in both hypotheses and con­ clusions, the converse implications hold also. (x) By (vii), C and D are of the form C = A X, D = B Y with X, Y + 0. We assumed the existence of C + D = (A + X ) + (B + Y). Hence by (i), (ii),

P R O P E R T I E S OF T H E EQ U IV A L EN C E CL ASS ES

47

(X + A ) + (B + Y) = X + {A + (B + Y ) ) = X + ( (A + B) + Y ) = = X + (Y+ (A

+ B)) = (A + B) + (X + Y)

whence A + B exists, and A + B ^ C + B by (vii). (xi) By (ix), A < B and f — A f — B imply t = A + (f - A) < B + (f - A) £ B + (f - B) = f and f < t* which is impossible. Hence by Corollary 5 to Definition 6.3, A < B implies f — A > f — B. Interchanging A, B yields th at A > B implies f — A < f — B. Clearly A = B implies f — A = f — B. Hence A ^ B imply | — A $ f — B respectively. Since we have exhaustive disjunctions in both hypotheses and conclusions, the converse implica­ tions hold also. D efin it io n 6.6: Put 0 • A = 0. If (n — \ ) A has been defined, put nA = (n — I)A + A if {n — \ ) A + A exists. Otherwise nA is undefined. (Corollary : n{A + B) = nA + n B and (n + m)A = nA + mA (in each case, both sides exist if either side exists) - Ed.) T heorem 6.4: nA is defined for every n ^ 0 if and only if A = 6. P roof : The reverse implication is trivial. Conversely, let nA be defined for every n. Then suppose th at av • • •, an are elements of A such th at (av • • •, an)_L. Then (a± + • • • + an) e nA. Since (« + I)A exists, A ^ | — nA by Theorem 6.2(b). Now let x n be an inverse of a± + f- an , whence x n € (f — nA). Consequently there exists an+1 e A with an+1 ^ x n by the dual of Corollary 3, Definition 6.3. Since («! + ••• + «n)^n+i ^ («i + b «n)^n = 0, and (av • • •, an) ± , it follows th at (av • • •, an+1) J_. Moreover, av • • •, an+1 c A. Thus if ax is any element of A, we have an infinite sequence (a{; i = 1, 2, • • •) such th at at € A, (av • • •, a{)_|_ (i = 1, 2, • • •). Thus by Theorem 2.3 (a i = 1, 2, • • •) J_. We have also axr ^ • • •, whence ax = 0 by Theorem 3.8, and A = A a^ = 0. T heorem 6.5: Let A ^ 0 , B be given. Then there exist uniquely an integer n ^ 0 and a class B x < A such th at B = nA + B v P roof : We shall establish the existence indirectly. Assume the theorem false. It will be proved by induction th at nA ^ B for every n. This is obvious for n = 0. Suppose it true for n = m ^ 0. Then there exists B x such th at mA + B x = B by Theorem 6.3 (vii). Now B 1 ^i A, since otherwise the theorem is true. Hence by Theorem 6.3 (vii) there exists B 2 with B x = A + B 2. Thus mA + [A + B 2) = B, mA + A ^ B, whence (m + I)A is defined, and ^ B . Thus all nA, n ^ 0, are defined, and so A = 0 by Theorem 6.4, contrary to the hypothesis. This proves the existence.

PART I

48

•

C H A P T E R VI

Let B = n xA + B x = n 2A + B 2> B x < A, B 2 < A, n x ^ n 2. By sym m etry it suffices to consider n x < n 2. Hence there exists m ^ 0 with n2 = nx + 1 + m, and n 2A -f- B 2 = (n x + 1 + tn)A -j- B 2 = n xA -f- {A + mA -f- B 2), whence B x = A + mA + B 2 by Theorem 6.3 (vii). Thus B x ^ A, contrary to B x < A. Hence n x = n 2. Then B = nxA + B x = n 2A + B 2 = n xA + B 2 and B x = B 2 by Theorem 6.3 (vii). D e fin it io n 6.7: Let A ^ 0, B be given. The unique integer n ^ 0 such th at B is of the form B = nA + B v B x < A will be denoted by [B : A]. Thus B = [B : A ]A + B v D e f in it io n 6.8 : An element ^4 of is said to be minimal (A min) in case A > 0 and there exists no element B such th at A > B > 0. T heorem 6.6 : If A ^ 0 is not minimal, then there exists B ^ 0 such

th at 2B

A.

P ro o f : If A is not minimal, there exists B x with 0 < B x < A. By

Theorem 6.3 (viii), A m ay be expressed in the form B x + B 2, B 2 =£ 0. Since B x ^ 0, B 2 < A. Hence if B is defined as the smaller of B v B 2 (use Corollary 5 to Definition 6.3), we have, by Theorem 6.3 (vii), (x), A = B x -f~ B 2 ^ B -f- B = 2 5 . D e fin it io n 6.9: A minimal sequence (A*) of elements =£0 in JS? is one containing but one element A x which is minimal, or containing a denumerable infinitude of elements such th at 2A i+x ^ A t (i = 1, 2, • • •). Corollary : There exists a minimal sequence. At this point we fix attention on a minimal sequence (^4,) which will remain fixed throughout the ensuing discussion. L emma 6.6: For every A, B, C for which the symbols are defined,

(a) (b)

[A : C] + [B : C] ^ [A + B : C] ^ [A : C] + [B : C] + 1. [ A : B ] - [ B : C ] £ [A : C] < ([A : B] + 1)([B : C] + 1).

P roof :

(a) P u t p = [A : C], q = [B : C]. Hence A = p C + A l9 B = qC + B Xt A x < C, B x < C. Therefore A + B = (^> + ^)C + (^4X+ B x). Now Ai + Bi = rC + ^42, where r ^ 0, ^42 < C. If r ^ 2, then ^4X+ A x < C, < C. Hence r = 0 or 1, and -4 + B = (/> + q)C + ^42, or A + B = (/> + £ + 1)C + A 2.

^ 2C, contrary to

P R O P E R T I E S OF T H E EQ U IV AL ENCE CL ASSE S

49

Thus [A + B : C] = p + q or p + q + 1; but

p+i i { P pXq q +i\ i p + i

+'•

whence (a) follows. (b) P u t p = [ A : B ] f q = [B : C]. Then A = p B + B lt

B = qC + Clt

B x < B, Cx < C,

and A = pqC -f (pCx + B x). But pCx + B x m ay be expressed in the form rC + C2, C2 < C, whence A = (pq + r)C + C2, and [A : C] = pq + r. Now r < p + q + 1. For suppose r p + q + 1; then r = [pC1 -f- B x : C] = [Q + • • • + C1 + B x : C] P

^P L C .iQ + l B ^ .Q + p

(by (a))

which is a contradiction. Consequently Pq ^ P q + r < pq + p + q + 1 = (p + l)(q + 1), and (b) follows. Corollary :

(a) (b)

If A ^ B and C # 0, then [4 : C] ^ [B : C], If 6 < B ^ A, then [ C : £ ] ^ [iC : A ].

P ro o f :

(a) There exists B x with B = A + B v Hence [A : C] + [Bx : C] ^ [£ : C], and [4 : C] ^ [B : C]. (b) P u t n = [ C : A]. Then C = nA + A v A x < A, whence C ^ n B + i4r Hence [C : A] = n ^ [C : B]. L emma 6.7: If A ^ 0, then l i m ^ ^ [A : A ^ = oo in the case when the minimal sequence (A{) is infinite. P r o o f : We shall prove first th at [A : A {] cannot be zero for every i. For suppose [A : A {] = 0 (i = 1,2, • • •). Then A = 0 • A t + B it B { < A it and A = B { < A it whence A < A { (i = 1, 2, • • •). By Lemma 6.6 (b), if H is any element of «£?, [H : A i+1] ^ [H : A & A , : A M ] ^ 2[H : A 0, < + oo. (If (.4*) consists of one element A x min, we mean by lim,-^^ the value at i = 1). P r o o f:

I. Let A { = A x be minimal. Then A x ^ 0 and B is of the form [B : A { \ A X + B v B x < A v Hence B x = 0, and B = [B : A t]Av Now [£ : A{\ 0 sinceB ^ 0 . Similarly A = [A : A X] AV whence [.4 : A x] 0. [B : A {] [B : A x] Hence U m ---------- = ------------ exists and has the desired properties. i^oo [A :A^ [ A: A,] F F II. Let ( A^ be infinite and minimal. By Lemma 6.7, [B : A {] and [A : A ^ have the limit + oo, whence each is zero for at most a finite number of values of i. By Lemma 6.6, [B : A i+j]

{[B : A ,] + 1 )([^ ,.: A i+i] + 1),

[A : A i+i] Sg [A : A t][A{ : A i+i]

(i, j =

1, 2, • • •),

whence [B : A i+i] ^ [B : A i] + 1 _ [At : A t+i] + 1 [A:AM ] ~

[A : A t]

'

+ {:

^ [B : A,] + 1

i1 [A AM ) \ [At: A i+i]/

[A : A {] [A : A ^

[A, : A i+j]

\

V!

Therefore r - [B : A i+i] [B : A,] + 1 lim ------------- s ------------------ » j-+oo [A : A i+j] [A : At]

K9 v "

P R O P E R T IE S OF THE EQUIVALENCE CLASSES

51

that is _ [ B : A h] ^ [ B : A f] + 1 lim ■ ■ j jfc— >00 [^4 : ^4^] [y4 : so that _ [BrilJ + l lim ----------S hm ----------------fc->oo [^4 : A A i=^o [A : A t] y

— l i m --------- B implies (A : C) > (B : C). P roof:

(a) This is obvious, since [A : AA![A : A A = I for i sufficiently large. (b) W

(A : B) = Km ^ ' Ai i = Km * = {B : 4 ) " 1. V 1 i^oo [B : [C :4,] ^ [ B - . A d i ^ l C i A d V n ’

(d) I. Let A x be minimal, and put p = [A \ A x], q = [B : A{]> s = [C : Aj J. Then A = p A lt B = qAv C = sA v Thus A + B = (p + q)Av and (A+ B:C) =

[A -|~ B i A jl

PA~ Q

[C: A-jA

s

T

lJ = V -^ *

P

Q

s

s

= — + — = (-4 : C) + (B : C).

P A R T I • C H A P T E R VI

52

II.

Let (^4*) be infinite (i = 1, 2, • • •). By Lemma 6.6,

[A : A {] + [B : A {] ^ [A + B : A J ^ [A : A t] + If we now divide through by [C : A has a limit, and we have

[B : A (B : C).

(A :C) =

D e f in it io n 6.11: We define

(2)

D(a) = (

10

:™

. a # ®’

if a = 0.

T h eo r em 6.9:

(i) (ii) (iii) (iii)'

D(a) ^ 0, D{a) ^ 1, D(0) = 0, D{ 1) = 1. D(a + b) + D(ab) = D(a) + D(b). b imphes D(a) = D(b). c > a implies D(c) > D(a).

(iii)"

a ~ b are equivalent to D(a) —D(b) respectively.

P r o o f:

(i) These properties are evident. (ii) We shall first prove that ab = 0 implies D(a + b) = D(a) + D(b). If either a — 0 or b = 0 the statement is obvious. In the contrary case we argue as follows. Since ab = 0, A a+h = A a+ A b. Hence (Aa+b : t) = = (Aa : f) + (^6 : t) by Theorem 6.8 (d), and D(a + b) —D(a) + D(b). Now let us consider the general case. If x is an inverse ofab in bt then ab + x = bt ax — abx — 0. Hence D(ab) + D ( x ) = D ( b ) ; but a + 6 = a + a6 + x = a + a;, whence D(a) + D(x) = D(a + x) = D(a + b), and we obtain (ii) by elimination of D{x). (iii) Now a — 0 imphes b ~ a = 0, whence b = 0 and D(a) = D(b) = 0. The same is true if b = 0. If a, b =£ 0, then D(a) = (Aa : f) = (Ab : f) = = D(b). (iii)' If c > a, there exists an inverse b of a in c, so that c = a + b,

P R O P E R T I E S OF T H E EQU IV ALEN CE CLAS SES

53

ab = 0, b ^ 0. Hence D(c) = D(a) + D(b) > D(a), since Z)(6) = (^46 : f) > 0. (iii)" Suppose a < b. Then there exists a' with a ^ a r < b , and D(a) = D(a') < Z)(6) by (iii), (iii)'. Because of Theorem 6.1(a), inter­ changing a, b shows th at a > 6 implies D(a) > D(b). Clearly a ~ b implies D(a) = D(b) by (iii). So a ^ b imply D(a) « D(b) respectively. Since we have exhaustive disjunctions in both hypotheses and conclusions, the converse implications hold also.

P A R T I • C H A P T E R VII

Dimensionality We first prove a strengthened form of the additivity of D{a) (Theorem 6.9 (ii)): L emma 7.1: (ax, • • *, an) _L implies D(ax + ••• + « „ ) = D(a1) + • • • + D(an). P roof : For n = 1 this is trivial. Suppose the lemma holds for n = m.

If K * * •, • • •, am) J_, and (ax H whence by Theorem 6.9 (ii) D ( al

+

’ • • +

am+1 )

==

D { al

—

& (al)

+ +

* * * + * * * +

^m ) +

D ( am)

b am)

and the lemma holds for n = m + 1. L emma 7.2: Let a be any element of L and (at ; i = 1, 2, • • •) a sequence, finite or denumerably infinite such that ID W

£D(a);

i

then there exists a sequence (a*; i — 1, 2, • • •)_]_ such th at (i = 1, 2, • • •). P roof : The desired sequence will be constructed by induction. Assume th at av • • •, have been defined, and th at (a'x, • • •, an_x)J_, ^ a (* = 1, • • •,n — 1). For w = 1 this is vacuously true. We proceed now to the construction of an. Since a\ ^ a (i = 1, • • •, n — 1), «£ + ••• + ^ a. Hence there exists an inverse x n of + • • • + an_x in a. Then K + • • • + *'n-i)x n = 0, and therefore (a^, • • •, s n) J_; also «£ + ••• + an_x + x n = a. Hence by Lemma 7.1, D(a) = D(ax) + • • • + D(a'n_x) + D(xn) — D ( al)

+

' ** +

D ( an- 1 )

+

D ( x n)-

B ut by hypothesis D(a) ^ 2 D (ai) ^ D M i

{54]

+ * ' • + DK )>

DIMENSIONALITY

55

whence D(an) ^ D(xn), and an < x n by Theorem 6.9 (iii)". Thus there exists an such th at an ~ an ^>xn, and («; + ••• + an^ ) a n £ &

+ ••• + a'n-i)xn = 0,

whence (a*; i = 1, • • *, n)_L- In this manner we obtain the sequence (a'; i = 1, 2, • • •)> where a (i = 1, 2, • • •)• Since (aj, • • •, an)_L (n = 1, 2, • • •), we have (a^; i — 1, 2, • • -)_L by Theorem 2.3. T heorem 7.1: If (at; i = 1, 2, • • •) is an independent sequence, finite or denumerably infinite, then

(1)

£ ( ! > < ) = 2 £>K). i

*

P roof : If (1) holds for each a{ ^ 0, it holds when this condition is not satisfied, since introduction of zero terms in either member of (1) leaves the equation unaltered. Hence we m ay assume ^ 0, and thus D{at) > 0. If the sequence (at) is finite, (1) follows from Lemma 7.1. Let (at) be infinite. Since (a*; i = 1, 2, • • *)JL, we have

(«i H

f-«») 2 a(6) = Z)(«' + a?) = D{a') + D{x) = Z)(a) + £(*), and Z)(ar) = D(b) — D(a) = /3 — a. Hence (/? — a) eZl. (iii) Let ax be any element such th at D(a1) = 0. Then there exists a sequence 0 ^ a x < a2 < • • • < z such th at lim^QQ a n = z. Since 0 ^ a n < a n+1 ^ 1 (n = 1, 2, • • •), we have the existence of (a^; n = 1, 2, • • •) for which a„ < a ' ^ a n+1, «■'„ e d

(» = 1, 2, • • •).

Thus a( < otg < • • *, and l i m ^ ^ a ^ = z. Hence by property (iii), z € A'. Consequently A' contains every z with 0 ^ z ^ 1, and thus coincides with the interval. This establishes Case oo. T heorem 7.3: The range A must be one of the sets A v A 2, • * •, A ^ defined in Lemma 7.4. We say accordingly th at Case 1,2, • • *, oo respectively holds for L. P roof : This results by combining Lemmas 7.3 and 7.4. It is easy to see now, by using the axiomatic treatm ent of projective geometry of G. Birkhoff or that of K. Menger, th at Case N corresponds to iV-dimensional, projective geometry (N = 1, 2, • • •). This will be discussed somewhat later. Case oo, however, corresponds to an entirely new system, to the study of which the subsequent chapters will be devoted. We have obtained the dimension function D(a) by a constructive process, in the course of which the principle of duality was not always observed, and which contained m any details of which no trace appears in the properties of D(a) ultim ately obtained. I t is essential therefore to show th at D(a) can be characterized by these properties alone, in­ dependently of its construction. D efin it io n 7.2: A real-valued function D'(a) whose domain is L is called an (unnormalized) dimension function in case

(a) (b)

the range of D ’(a) has either an upper bound or a lower bound, D'(a + b) + D'(ab) = D'(a) + D'(b).

T heorem 7.4: Every function

D'(a) = y xD{a) + y 2(yv y 2 real numbers) is a dimension function, and every dimension function is of this form.

DIMENSIONA L IT Y

59

We observe first th at if D"(a) is a dimension function, then every D'(a) = y 1D"(a) + y 2(yi> 72 rea* numbers) is also a dimension function. Now D(a) is a dimension function, since it satisfies (a) because D(a) ^ 0 and (b) by Theorem 6 .9 (ii). Hence every D'(a) of the theorem is also a dimension function. We need then prove only the converse, i.e., th at every dimension function is of the form y xD(a) + y 2. If a, x are inverses, then (b) yields P roof:

D ’(a) + D f(x) = D'(a + x) + D'(ax) = D'( 1) + D ’(0), and D ’(a) = D ’(1) + D'(0) - D ’(z). So if b, then D'(a) = D ’(b). Thus by Theorem 6.9 (iii)", D(a) = D(b) implies D'(a) = D'(b), and so D'(a) is a function of D(a): D ’(a) = f(D(a)), where f(x) has the domain A. Assume x, y e A, x y 1. Then by Theorem 6.8 (d), (e) there exist a, b with D(a) — x, D(b) = y, ab = 0. Then D(a + b) = D(a) + D(b) = x + y, so th at /(* + V) + /(0) = f(D(a + b)) + f(D(ab)) = D ' ( a + b) + D'(ab) = D'(a) + D'(b) = /(*) + /(, C a re b o t h in v e r s e s o f e le m e n t a s u c h t h a t b = c, t h e n b = c. P r o o f : By symmetry, we may assume b ^ C. Then P roof: L em m a

c = (b u a) n c = b u (a n c)

a g iv e n

(by IV)

== b u 0 = b. D e f i n i t i o n 3.3: If aif a,- are two elements in a complemented modular lattice L such th at a* n cq = 0, we define L tj as the class of all inverses ha of a,- in a* u a*. L e m m a 3.4: Let L — be the lattice of principal right ideals of a regular ring 31. Let ait Ct,- e L with a* n a ; = (0), and let ^ ^ be idempotents such th at a* = (£t)r , a, = (^)r, = 0 (cf. Lemma 3.2). Then the set L {j is precisely the set of all (et- — v ) r> with v e 31, and

(5)

ejv = vei = v.

96

P A R T II

•

C H A P T E R III

Moreover, each inverse h{j determines uniquely the element v € 9ft associated with it. P roof : Let b^-cL^-. Since c= a;- u h^, there exist v e d jy s e b nwith e{ = v + s. Now (s)r c: bw. But n

(s ) r

c

u

b ti

=

af u a*;

also cty u (s)r => a,-. Since v + s = (s)r , whence a* = (e{)r c: a* u (s)r , we have a;- u (s)r a* u a,-. Hence a;- u (s)r = at- u a,-. Since a;- n (s)r = (0), we see that (s)r c= b# is an inverse of a,- in a* u a,-; by Lemma 3.3 bi* = (s)r = (e» — v)r- Inasmuch as v e a 3-, e^v = v. Since ^ — t; c b # , — t/)(l — eijebij-; but (et - v) (I - et) = - v(l -

e a,-,

whence —v(l — e^) = 0, and i; = i/g,-. Thus (5) is established. Conversely, let us consider bti = (e{ — v)r, where v e 9i satisfies (5); we must show th at b# e L {j. Suppose a; e a*nb#. Then x = a; = v)z, with z € Hence a; = — v)z = — vz = — vz, whence x — (ei—v)z = —vz, e{z = 0, and therefore x = — vz — — vei - z = — v - e i z = 0. Thus a* n = (0).Now since vect*, a,- u h^ = (*,)r u

- v)r ci a< u a*.

But (^)r u ( ^ —v)r a,-, and since >2: = etz = vz + (ei~v)z e (^)r u — v)t for every 2 e a*, we have (^)r u ( ^ —v)r => a,-. Thus a,- u b i3- = (e;)r u (^ —v)r = a* u a*, and b# € Finally, let v, v' satisfy (5) and suppose — v)r = — i>')r. Then since e* — v, et — v' e bw, it follows that v — v' = (Ci — v,) — (ei — v) chtj. But v — v' € (Xjt since v, v' e dj by (5). Thus v — v' = 0, i.e., v = 1/', and v is unique. The lemma just proved establishes a one-to-one correspondence ha = (e{ — v)r between all elements v e 9t subject to the conditions (5) and all inverses b# of a* in a* u a?-. The notations dif a;- were used since in applying the lemma we shall be dealing with independent systems (cti, • * •, an) of which ait dj are members. D efin itio n 3.4: Three elements a, b, C of a complemented modular lattice L are said to be in the relation ((a, b, Of€) in case a u b = b u c = c u a, a n b = b n c = c n a = 0, i.e., in case each element is an axis of perspectivity of the other two in their join.

O R D E R OF A L A T T I C E

97

Lemma 3.5: Let L = R m be the lattice of principal right ideals of a regular ring 9t. Let aif cq e L with a* n a,- — (0), ax* of 91 generate the involutoric dual-automorphism a -> Cl' of R m . Then the

ISOMORPHISM THEOREMS

115

following properties are equivalent: (a) (a') (b) (c) (d)

a n a ' = (0) for every a € R m; a u d' = 91 for every a e R m; for every a e 91 there exists 2 € 91 with a = za*a; a*a = 0 implies a = 0 ; for every a e R there exists a Hermitian idempotent e € 91 such th a t (a)r = (e)r.

Moreover, in (d) the Hermitian idempotent e is uniquely determined by (a)r = (e)r. P ro o f: The equivalence of (a) and (a') is immediate since (a n a ' ) ' = a ' u a " = a u a', (a u a')' = a n a' (because a -* a' is involutoric), and since (0)' = 3ft, 9?' = (0). We shall now establish the equivalences (a) (b), (a) (c), (b) (d). Now a n a ' = (0) means th at z e d , a' implies z — 0, i.e., z e (ax; x), (y : a*y = 0) implies 2 = 0, i.e., a*ax = 0 implies ax = 0. This condition is equivalent to (x; a*ax = 0) cz (x; ax = 0), hence to (a * a ){a (a)tr, and therefore to (a*a)t zd (a)t by Lemma 2.1 (1). But this is manifestly equivalent to a e (a*a)t for every a e 9ft, i.e., to (b). Hence (a) (b). Now a n a' = (0) for every a e R m means that a cz a' implies a = (0). For, if a n a' = (0), a cz a', then a = a n a' c (0); conversely a n a ' cz a u a' = ( a n a ' ) ' , whence a n a ' = (0). Hence (a) means th at (a)r cz (;y ; a*y — 0) imphes (a)r = (0), i.e., imphes a = 0. Hence (a) is equivalent to this property: a*ax = 0 for every x e 91 imphes a = 0, i.e., to (c), since a*a = 0 if and only if a*ax = 0 for every x e 91. Thus (a) ?± (c). Let us assume (b), viz., that every a e 31 is of the form a = za*a, z e SR. Define e = za*. Then e* = a**z* — az*; hence e* = a • z* = za*a • z* = za* • az* = ee*, and e = e** = (ee*)* = e**e* = ee* = e*; moreover e2 = e • e = e • = e. Thus £ is a Hermitian idempotent. Now a = za* • a = ea, whence a e (e)r , (a)r cz (e)r . B ut e = az*, whence e e (a)r , (e)r a (a)r . Therefore (e)r = (a)r , and (d) is established. Con­ versely, let (d) hold. Then (a)r = (e)r, with e idempotent and Hermitian. Consequently, e is of the form e = ax, xeSR and a — ea; hence e = e* = x*a*, a = ea = x*a*a, whence z = x* yields a = za*a. Thus (b) (d). To establish the uniqueness of e in (d), let (e)r — (f)T, with e, f Herm itian idempotents. Then fe = e, ef = f, whence f = f* = (ef)* = f*e* = fe = e.

116

P A R T II

•

C H A P T E R IV

For the purposes of the following theorem, in which we analyze the implications of conditions (b), (c), (d) of Theorem 4.5, we drop the as­ sumption of the regularity of 91. 4.6: Let 91 be a ring (with unit) and let x -+x* be an involutoric anti-automorphism of SR (i.e., x** = x for every xeSR). Then conditions (b) and (d) separately imply that 9t is regular; (c) implies that 9i is semi-simple (cf. Chapter II, Appendix 1 (4)), but not that 9? is regular. P r o o f : Since condition (d) is clearly a strengthened form of Theorem 2.2 (/?), it obviously implies regularity. We prove next that (b) implies regularity. Now if a = za* a, then a* = a*a**z* — a*az*, whence az* = za*a • z* — z • a*az* = za*. Thereforea = za* • a = az* •a = axa with x = z*, and we have established condition (y) of Theorem 2.2 and hence regularity. To prove that (c) implies semi-simplicity, we repeat essentially the argument given in Chapter II, Appendix 2 (I). The contrapositive of (c) is that a ^ 0 implies a*a ^ 0. Hence if a* = a, then a ^ 0 implies a2 ^ 0, and moreover (a2)* = (a*)2 = a2. Therefore by iteration, a ^ 0 , a* = a implies a{2n) = (• • • ((a2)2) • • *)2 ^ 0whence am ^ 0 forevery n and m 5^ 2n. Thus a ^ 0, a* = a impliesam ^ 0 for everym. But (aa*)* = a** a* = aa*} whence we see that a ^ 0 implies a* ^ 0, aa* ^ 0, (aa*)m ^ 0. Now if a is a right ideal t^(0), we may select a e a with a ^ 0; then aa* e a, (aa*)m €am, (aa*)m ^ 0, whence am ^ (0). Thus a is not nilpotent, and by definition is semi-simple. To show that (c) does not imply regularity, let 91 be the ring of all bounded operators in a Hilbert space § (cf. Chapter II, Appendix 2), and define a* to be the Hermitian adjoint of a. Then a*a — 0 implies a = 0; nevertheless 9t is not regular (cf. loc. cit., (II)). T heorem

P A R T II • C H A P T E R V

Projective Isomorphisms in a Complemented Modular Lattice In what follows, until the contrary is stated, L is supposed to be a given complemented, modular lattice of order n (n ^ 4); for the present weshall assume a homogeneous basis (a*; i = I, • • *, n) to be fixed in L. The theory in this chapter is preliminary to the introduction of a regular ring 91 associated with L such that L is isomorphic to the lattice of all principal right ideals in 91; this result has been mentioned at the beginning of Chapter IV. The essential result of this chapter is th at contained in Theorem 5.1; Lemmas 5.4—5.10 are preparatory for it. Unless otherwise indicated, indices i, j, k, • • • vary over the range (1, • • -,n). Lemma 5.1: If i ^ y ^ k i, bt-3- e L^, b3fc e L j k, and if bik = (£>;; u b,-*) n (a, vj a*), then bik e L ik. P r o o f : We have a* u brt = a* u ((b« u bjk) n (a< u a*)) = (a* u bn u bjk) n (a,- u a*)

(by IV)

= (a* u a, u bH) n (a{ u ak) = (a* u a, u a,-) n (a,- u a*) = 5iU a*. Moreover, since (a*-, ajt afc)JL, it follows from P art I, Lemma 2.1, th at (hijf cq, ak)_L, and hence by a second application of this lemma, th at (fhi* 'bik* 6*) _L- Thus (&« u &**) n f l ^ O , and a* n bf* = (bw u biA:) n ( a z- u a*) n afc = (bw u b3fc) n afc -

0.

Consequently bik is an inverse of ak in a* u afc, i.e., bik e L ik. D e fin it io n 5.1: If i ^ j ^ j ^ k ^ i t 5 .. 6 L ij} bjlc c L jk> then we define b{j 0 bjk = (bt-3 cj b#Jfc) o (cq cj &&). Corollary : If b« e L ijf bjk e L jkt then bw 0 bjk e L ik. Lemma 5.2: The operation 0 is associative, i.e., for i, /, k, I distinct integers, b^ € L^jf bjk € L jkf bki e L k\ we have (1)

(fr*j 0 bj k) 0 bki — bjj 0 [117]

(b3fc 0 bfci).

P A R T II • C H A P T E R V

118 P roof:

The left member of (1) = {(b^ u b ^ a * u ak u at) u bfcI} (at- u a*) (by IV)

= ( B « u b , t u b w)(a< u a , ) = the right member of (1).

D e f i n i t i o n 5.2: A system (cu \ i, / = 1, • • *, n, i ^ j) of axes of per­ spectivity for the homogeneous basis (a*; i = 1, • • •, n) (i.e., c e L ijf L H) is normalized in case

(2)

Cij

(3)

Cik — tij ® Cjk

Cji

(* + i) i).

The combined system (a*, CtJ-; i ^ j) is a normalized frame for L in case (2) and (3) hold; if (2), (3) are not required to hold, this system is simply called a frame. L e m m a 5.3: There exists a normalized system (ctJ; i ^ j ) for the basis (at); the elements ca € L iv L u (i = 2, • • •, n) may be chosen arbitrarily and the remaining members are then uniquely determined. P ro o f: The existence of the ctl as common inverses of av a* in cq u at(i = 2, • • *, n) is obvious since av If (2) is to hold, we must define C1;. = Cn (j = 2, • • *, n), and if (3) is to hold, we m ust define (4)

C„

= ctl Cj, = (ctl u c1}.) n (a, u a,)

(i, j ^ 1, i

/)•

Equation (2) is now evidently satisfied. Since ctl e L {1 (i = 2, • • •, n), l xj € L ±j (j = 2, • • •, n), it follows from Lemma 5.1 th at € L {j (i, j ^ 1, i ^ j). Hence c„- e L {j for every i, j with i ^ j, whence cif = l H e L H also, and c i s an axis of perspectivity for ait a3-. It remains to verify (3). If / = 1, (3) follows from (4). Suppose then th at / ^ 1. First, let i — 1, whence k ^ 1, j. Then Cij ® Cjk — CU ® (C,1 ® Cifc) — cji ® (c* ® C^) = fei u W n v cu ) n (a,- u a*)]} n (aj u a*) = (2* uCw) ^ (Cn u % u a*) n (ax u a*)

(by IV)

= (en u t i t ) n (ax u o , u a*) n (ax u a*) = (2,i u cw) n (a, u a*) = Cit n ^ ( f li u a*)

(by IV)

= Cut r> 0 = cu . Thus (3) holds for i — 1. Next, let k = 1, whence i ^ 1, /. Then an analysis similar to the one just made shows th at (3) holds for this case

PROJECTIVE ISOMORPHISMS

119

also. Finally, let us consider the general case where 1, Then C'j 0 £jJc =

(^il & ^1j) 0 tjk 0 (Cij 0 cjk)

i, j, k

are distinct.

(by Lemma 5.2)

= C C a', i ^ j ) will be a fixed normalized frame throughout the following discussion until the contrary is asserted. Throughout the remainder of the chapter we shall be dealing with sequences of m integers i p , j p> • • • (p = 1, • • *, m), m < n ; we make the convention th at in each of these sequences the m integers shall be distinct and shall belong to the range (1, • • •, n). L em m a 5.4: If ( i v • • • , i m ) , ( j v • • • , j m) are such th at i p = j p except for one value p of p, then m m

(5)

lu % ~ lu %

/>=1

P ro o f:

Since a

= Sy

(mod V P '

p=1

(aip; p

=

1, • • •,

9 9

p ^ p)

m,

= Du ( d y p = 1, • • *, m, p ^ p), and since (a, dt_, a,_)_L, (mod ii-j-) implies by the proof of Theorem 3.5 in P art I, a u

a u a,-_

(mod Ci j ). This is precisely (5). D

e

f i n

i t i o

n

5 .3 :

Let

( i lt

• • •, * m ) ,

( / i>

’

’

% 7 « .)

be given. If

ip =

jp

(p = 1 • • w), then we define P i* 1 | as the identical automorphism Vh’ - ’W of L(0, 2y&* )• ^ ip = i p except for one value p of p, then we define

9=1 P \} L(0,

P

*m) as the perspective isomorphism defined by (5) of the lattices

L(0, & jp) the axisct _,_. (Cf. P art I, Definition 3.4.) p=i 99 L e m m a 5.5: Let (iv • • •, i m), (jlt • • •, j m) be such th at i p = j p (p = 1, • • •, m), or such th at i p — j p except for p — p. (a) If I — 1, • • *, m and plt •••, pi are distinct integers in the set (1, • • •, m )f CLi p ) ,

/>=i

then P j .1

) coincides with Jm/

VI

A= 1

A

P IV 1 %9l\ for 'J Pi * Jp j

the elements

120

P A R T II

(b)

If u

CHAPTER V

•

then P lm\ maps c,• ,• on c7 7 . X h '" 1 m J 9 9 9V P r o o f: For the case i p — j p (p = I, • • •, m) , both (a) and (b) are obvious. Suppose then ip =£ (a) Suppose i p ^ j p for p = p ^ rA (A = 1, • • •, Z). Then *Ta = / Ta i ••• i \ m for A — 1, • • •, /, whence P 1 m is a perspectivity of h * ' ‘ 1-mJ p=i 9 g,

(

m

and

^ p=l

9

TA

a=i

, and therefore is TA

)• Suppose i ^ / A=1_

A

I

cq a=i

i

the identity on L(0, i

I

which are both ^

for p = p — t j . Then

TA

= j t^ except for A = A; both our transformations P are perspectivities A

with the same axes c,_ 7_ = V

p

Since both domain and range of the

*t a ’ t a

b

second P are contained in those of the first P, respectively, we may conclude th at our two transformations P coincide there. (b) Suppose i p ^ j p for p = p= ju. We must then prove (mod

i-e*>

(mod

u v * = v , u =

= =

u

But this holds since £v v } n

(% u av )]

(a,-_ u a v u ct_ ,_) n (c*_

u s l i p i-p u

u a~0 n h

£v V

=

£v V

u

u c,_ v )

(by (b y I V )

’p u £W S

V

(s in c e

*«■ =

? -)•

w m and since c* t(r ^ 2 U a, , i,■ ip j and i an = /_, whence L

* is

left invariant, i.e., is transformed into c,-■'/i-'o,■ . L em m a 5.6: Let i, j, k, h be distinct. Then

(the operations in a product being applied in the order written). Consider u ^ cq vj cq. P \ J . | carries u into V ^ afc u a,- with /h \k V u u cifc = t) u c^; PI I carries b into tv ^ cq u a* with D u C j ^ t D u cjh. P ro o f:

PROJECTIVE ISOMORPHISMS

121

On the other hand PL. M carries u into t)' gS a< u aA with u u C,A = (i h\ ^ ' t)' u c,A; P L A carries b' into tb' ^ ak u aA w ith b' u i ik = to' u lik. Hence u u cik u cjh = b u cik uC,A = to u l ik u c jA; similarly u u c l t u c ,A= to' u ciku cjh, whence to u c i t u c ,A= tt»'ucf t u c tt. Moreover, to, to' aAu aA. Since (a,, a,-, ak, aA)_L, and since cik e L ik, cjhe L jh, P art I, Theorem 2.8 yields (cik, ljh, ak, aA)_L, whence (a* u ah) n (cik u c,A) = 0. Therefore t o = t o u O = t o u [(a* u aA) n (cik u c#*)] = (to u cik u cjh) n (a* u aA)

(by IV)

= (to' u cik u cjH) n (aAu aA) = to' u [(aAu aA) n (cik u c,A)]

(by IV)

= to' u 0 = to', i.e., to = to'. This establishes (6). C o r o l l a r y : If i, j, k, h are distinct, then

p(i;:)p(*i)p(‘J)p(:*)-

(a uckh = t v v c**.

= tv u

u c**. Then

(u u cjk u cM) n (a< u a, u aA) = (to u crt u c**) n (a* u a, u aA); since u ^ a* u a;-, ft)

a* u a^, IV applies and we have

u u [(crt u e») n (5i u i j U aA)] = to u [(cjA u c**) n

(a, a,-u

aA u)].

Since cjk u cM ^ a,- u a* u aA, (c« u cAA) n (at- u a,- u aA) = (cjk u cAA) n = (cik u cAA) n (a,

(a,

u aAu aA)

u aA) = cjA,

n

122

P A R T II • C H A P T E R V

the last equality but one holding by P art I, Theorem 2.5, since (a*, a,-, 5*, a*) J_, and the last equality holding by (3). Thus u u c ft = l t ) u cjh, and tv is the transform of u under P ^({. 1) This establishes (7). L emma 5.8: If i, j, k are distinct, then

(8)

V = p { [

/4

4 \ r J 4 T) 4 \ r , / 4 T)

4 t)

\

,«)=H“,ity

/ 4 W- 1)4 W - 1)\

by the induction hypothesis. Suppose the length of A' to be co. Now A' has v1 = 2 ; but A f has v° = = 2 (since for zl we assumed v1 = 1 and th at the numbers v1, • * *, v10 = v° assumed alternately the values 1, 2 ; A \ A have obviously the same v° — v0*). Hence for A', v° = v1 and (11) holds by the result in (A). Thus (11) holds in any case. Moreover,

p(;:9

lu i2

L(0, 2 u ^ ) '

By combining (as in the proof for case (B)) (18) and (19) we obtain 0 = Identity. This discussion shows th at the cases remaining to be treated are those in which r' ^ r, r ± 1 (mod co) implies i ^ i^ and similarly 4 T> # Let us assume that v1 = 1 (since the case v1 = 2 follows by sym m etry). By (17) we have 4 1J = 4 4) = 4 ?) (since is 1 f°r * odd and is 2 for r even). Thus we must have 1 = 7 or 7 ± 1 (mod co), i.e., co m ust be a divisor of 5 or 6 or 7. But we had co = 6, 12, 18, • • •, whence co = 6. For this case (10) m ay be written by (15) in the form

^ p(i!)pC ’ ' ' ’" '" ‘ ’'I

(h '''

where for each r = 1, • • •, co, ^ r-1) ^ ^ r) occurs at most once and for each r ' = 1, • • •, co', ^ ilT,) occurs at most once, then •(0 -1 ) . . . -( « -l)v

k - - - d

W*

J

= P (h---

... P

P ro o f : W e m ay w rite (23) in the form

(h • • •

( i r{ 1} ■■■> ir 1}\ . . . jOj>) \

U .

v r - 1’ • • • / r - 1’/

" V i-

/y(“ )

U ...

7 \ -

-

\

J

since clearly i„ = j„, t ’"1 = /*"'*. Thus Lemma 5.10 applies, yielding

k - '- d

L r

/ p //r

•••/r

p ( h • • •/;\

\

Ir-"- -/?-")'' I. ■J "

which may be written also in the form (24). T heorem

(P

T,

...

y'

5.1: There exists a system of transformations

;V

i P = l >--- , n (p = 1, • • ’ , m), i p =£ i a , j p ^

for

p zjk a, m = 1, • • • , « — 1 j

128

P A R T II

•

CHAPTER V

with the following properties: (i " ' i \

(a) PI }

.mJ is a projective isomorphism of L(0,

m v= l

V I ***

m

)andL(0, J l A ' )* V

(b) If I = 1, • • • , m and r v • • • , r t are distinct, then P ^*1 cides on L(0,

p a=i

Ta

V

coin­

(*-Ti V T j ’ * * I tJ

(c) If (I, a = 1, • • •, m, ft ^ a then P ^ J

(d)

v= l

tm) maps ci/i(