The mathematical apparatus for quantum theories; based on the theory of boolean lattices

The algebra of Boolean lattices looks like the algebra of sets, because we have there the operations of addition, multiplication and tation, obeying just the same formal rules, as in the

Die Gruncllehren cler

mathematischen Wissenschaften

in Einzeldarstellungen mit besonderer Beriicksichtigung

der Anwendungsgebiete

Band 129

Herausgegeben von

J L Doob E Heinz· F Hirzebruch E Hopf

H Hopf W Maak S MacLane

W Magnus· D Mumford· F K Schmidt· K Stein

GeJchajtsjiihrende HerauJgeber

B Eckmann und B L van der Waerden

The Mathematical Apparatus for Quantum -Theories

Based on the Theory of Boolean Lattices

Otton Martin Nikodym

Springer-Verlag New York Inc 1966

Geschaftsfilhrende Herausgeber:

Prof Dr B Eckmann Eidgenossische Techuische Hochschule Ziirich Prof Dr B L van der Waerden Mathematisches Institut der Universitat Ziirich

ISBN-13: 978-3-642-46032-6 e-ISBN-13: 978-3-642-46030-2 DOl: 10.1007/978-3-642-46030-2

All rights reserved, especially that of translation into foreign languages

It is also forbidden to reproduce this book, either whole or in part, by photomechanical means

(photostat, microfilm and/or microcard or any other means)

without written permission from the Publishers

© 1966 by Springer-Verlag New York Inc

Softcover reprint of the hardcover 1 st edition 1966

Library of Congress Catalog Card Number 66·27977

Tide No.5 I 12

Dedicated to my wife

Dr Stanislawa Nikodym

The theory exhibited in this paper has been simplified, generalized and applied to several items of the theory of maximal normal operators

in Hilbert-space, especially to the theory of multiplicity of the continuous spectrum and to permutable normal operators, based on a special canonical representation of normal operators and on a general system

of coordinates in Hilbert-space, which is well adapted not only to the case of discontinuous spectrum, but also to the continuous one The normal operators, which can be roughly characterized as operators with orthogonal eigen-vectors and complex eigen-values, constitute a generalization of hermitean selfadjoint and of unitary operators

The importance of the methods, sketched in the mentioned paper, has been emphasized in the review in the "Zentralblatt fUr Mathematik",

by the physicist G LUDWIG 2 and later applied by him in his book

"Die Grundlagen der Quantenmechanik"3 The mentioned theory has

1 Annales de l'Institut HENRI POINCARE, tome XI, fasc II, pages 49-112 The paper constitutes the content of four lectures by the author: February 4,6, 11 and 13, (1947) at the Institut HENRI POINCARE in Paris

2 Bd.37, 1951, P.278/279

3 Berlin/Gottingen/Heidelberg: Springer-Verlag Die Grundlehren der matischen Wissenschaften 52 (1954), XII + 460 pp (see the footnote p 75)

Mathe-VIII Preface

later been simplified, generalized and applied in several papers by the author The present book can be considered as a systematic syn-thesis of them all, with suitable preparations, additions and precise proofs

It contains many new notions, as the notion of "trace", which are defined, studied, and applied

The author hopes that this book will be useful not only to physicists but also to mathematicians

Concerning the Boolean-lattice-approach, the following remarks are

in order: J v NEUMANN has found interesting relations between the logic of propositions and some behaviour of projectors in the Hilbert-space Now, if we introduce with M H STONE suitable and simple operations on closed subspaces of the Hilbert-space, we can perceive that just the Boolean lattices made out of closed subspaces constitute the suitable, useful translation of the relations mentioned, found by

V NEUMANN, and that the Boolean lattices should be chosen as a convenient background for further developments

An other source can be found in the modern theory of set-function and of general, abstract integration and measure, created by DE LA VALLEE-POUSSIN, VITALI, HAHN, RADON and especially by M FRECHET who has generalized the LEBESGUEan theory to abstract sets and general denumerably additive, non-negative and bounded measure The above few sources have made it possible to construct a geo-metrical theory of selfadjoint operators and to extend it to normal operators

We mention that the original approach to the mathematical part

of the theory of quanta, based on matrices, is n.ot adequate, as has been shown by J V NEUMANN in his paper: [J reine angew Math 161 (1913)] The matrices have been replaced by operators in Hilbert-space (F RIEsz, J V NEUMANN, M H STONE)

Since we do not require that the reader be familiar with the modern abstract theories, we shall start with a sketch of the theory of Boolean lattices

The reader is supposed to be familiar with

1) basic properties of the structure of Hilbert-space, with basic properties of hermitean selfadjoint operators, the Hilbert spectral-theorem included,

2) with the theory of Lebesgue's measure and integration,

Preface IX 3) with basic notions of abstract topology and

4) with the notion of an ideal in a commutative ring The reader

is also supposed to know the necessity of discrimination between notions having different logical type e.g between a set and a set of sets

We apply the usual notations with the following novelty: we shall sometimes use a dot over a letter, say x, to emphasize that x is a variable,

e g f (x) means a function of the variable x, and f (a) means the value

of the function at the point a, the symbol A,i will mean the sequence

{AI' A 2 , ••• , All""} and n the sequence 1,2,3, , n, of

We are including the list of contents of these chapters

We give the list of references labelled with a fat parenthesis ( ) The list contains not only the papers which are directly applied in the text, but also all those papers, which have influenced the author with some useful ideas

Though the "Apparatus" is destinated for physicists, it does not contain direct applications to mathematical problems of physics The author intends to deal with them in subsequent papers or in another book

I am owing my thanks to Prof Dr HELMUT HASSE and to Prof Dr

B L VAN DER W AERDEN for their kind recommendation of my work

to the Springer-Verlag

I wish to thank the U.S.A.-Atomic Energy Commission and the U.S.A.-Office of Ordnance Research for their financial help in my research related to the book, and especially I am owing my thanks to the U S A National Science Foundation for support through several years I am owing special thanks to that institution whose grants have made possible the final composition of the book

In addition to that I express my thanks to the French "Fondation Nationale des Recherches Scientifiques," whose financial aid has made possible my research on the "Apparatus" in 1946-1948, and especially

to Professor ARNAUD DEN JOY who kindly arranged that financial aid

x Preface

But my most hearty thanks I am owing to my wife Dr Sl.AWA NIKODYM, (also a mathematician) whose help in composing the book, proof reading and typing was very great Without her efficient help it would have been impossible for me to compose the present book

STANI-Finally, I would like to thank the Springer-Verlag and the printers

fm a beautiful and very clear setting of a quite difficult text

Utica, N.Y USA,

List of chapters

A General tribes (Boolean lattices)

A 1 Special theorems on Boolean lattices

B Important auxiliaries

B 1 General theory of traces

C The tribe of figures on the plane

F Linear operators permutable with a projector 386

G Some double STIELTJES' and RADOX'S integrals 391

H Maximal normal operator and its canonical representation 402

J Operators N j(x) =df q; (x) j(i) for ordinary functions! 428

J 1 Operational calculus on general maximal normal operators 450

K Theorems on normal operators and on related canonical mapping 462

L Some classical theorems on normal and selfadjoint operators 502

M Multiplicity of spectrum of maximal normal operators 521

N Some items of operational calculus with application to the vent and spectrum of normal operators 560

resol-P Tribe of repartition of functions 586

P 1 Permutable normal operators '

Q Approximation of somata by complexes

Q 1 Vector fields on the tribe and their summation

Quasi-vectors and their summation

Summation of quasi-vectors in the separable and complete Hilbert-Hermite-space General orthogonal system of coordinates in the separable and complete Hilbert-Hermite-space

DIRAC'S Delta-function

Auxiliaries for a deeper study of summation of scalar fields Upper and lower (DARS)-summation of fields of real numbers

in a Boolean tribe in the absence of atoms

W 1 Upper and lower summation in the general case Complete

Chapter A

General tribes (Boolean lattices)

A.t This first chapter contains the fundamentals of the theory

by the author in (10) and in several subsequent papers by the author The algebra of Boolean lattices looks like the algebra of sets, because

we have there the operations of addition, multiplication and tation, obeying just the same formal rules, as in the theory of sets, but with the exception that the relation of belonging of an element

complemen-to a set, a E (x, is not considered at all We could roughly say, that the elements of the Boolean lattice are "sets without points", though the points may be even available

Usually people use the term "Boolean algebra", rather than the term Boolean lattice Now, since in the sequel we shall use not only finite operations, but also the infinite ones, the theory stops to be an algebra Since the infinite operations are defined as a kind of supremum and infimum of collections of elements, therefore the true basis of the theory is the notion of ordering (partial ordering) (6), (8) Therefore

we shall start with this notion and define the lattice as a kind of ordering

In agreement with RUSSELL'S and WHITEHEAD'S Principia Mathematicae (1), we define the ordering as a relation (correspondence, mapping),

satisfying certain conditions The relation (correspondence., mapping)

will be understood as a notion which is attached to a condition tional function) (1), involving two variables, similarly as the set is a

(proposi-notion attached to a condition (propositio~al function), with one variable only Thus e.g as the condition for numbers 3x - ~ x = 5 with the

2 A General tribes (Boolean lattices)

variable x generates the set {x 13 x - : = 5}, so the condition 3 x - y::;; 4

generates the relation {x, y 13 x - y ::;; 4}

We shall not consider the relation {x, ylw(, x, y.)} as the set of all ordered couples of numbers (a, b) satisfying the con-dition w( a, b.), but we shall stay with the original Russell's approach The statement: a is an element of 5, is usually written a E 5,

where 5 denotes the set {xlw( x )} Similarly the statement v ( a, b ),

where a, b are fixed elements, will be (optionally) written aRb, where

R denotes the relation {x, y 1 v ( x, y.)} with two variables x, y

The set of all elements a, satisfying the condition

"there exists b, such that aRb"

is called domain 0/ R We denote it by OR

The set of all elements b, satisfying the condition

"there exists a, such that aRb"

IS called range 0/ R We denote it by DR

The elements of the set must have the same logical type in order

to avoid logical antinomies, but in the relation aRb, the types of a

and b may be different

{x, y Iw(, x, y.)} differs from {y, xl w ( x, y )}

The relation is said to be empty, whenever there does not exist

a and b with w( a, b.)

Functions are considered as relations

A lattice will be considered as a special kind of ordering and the Boolean lattice as a special kind of lattice

We shall use the term "tribe" (Boolean tribe) to denote a Boolean lattice The term is borrowed from RENE DE POSSEL (11) The elements

of an ordering will be termed "somata" (sing soma); this term is borrowed from CARATHEODORY (7)

A tribe, if we consider finite operations only, can be reorganized into a commutative ring with unit (Stone's ring, Boolean algebra)

We shall state and prove several simple theorems The proofs will

be precise and even sometimes meticulous in order that the reader, who is not familiar with the topic, be acquainted with methods and not spend his time by completing proofs, if they were only sketchy

A part of the chapter will be devoted to some notions related to the notion of equality 0/ elements This notion is usually considered

as something trivial But we can notice that the equality may differ from the identity

E g In Hilbert-space, whose vectors are square-summable functions

/ (x), we have the "equality almost everywhere" of functions / (x);

hence not the identity

A General tribes (Boolean lattices) 3 This fact compels us to make some necessary remarks concerning the so-called "governing equality" and the role it plays

The said discussion of the notion of governing equality enables us

to treat the notion of a subtribe and supertribe of a given tribe with

more precision than in many textbooks

I t will enable us to introduce the precise notion of finitely genuine subtribe of a given tribe and some variations of this notion They will

be very important in the sequel

Since the tribe can be reorganized into a ring, we can speak of

ideals in a tribe The ideal generates a notion of equality, which

recon-structs the tribe into another one We shall devote to ideals a small part of the chapter

As the elements of a tribe behave like sets, we can introduce the notion of measure of somata, which in turn genlirates a special ideal and

also the notion of the distance between two somata The distance organizes,

under simple condition, the tribe into a metric space; hence it yields a topology In dealing with measure, and generally with functions, we shall put for more clarity a dot upon a letter, say x, to emphasize

that x is variable Thus f (i) is a function and f (x) the value of the

function at x For more subtile details concerning tribes we refer to

our two papers (12), (13)

The part A deals with fundamentals only; more special theorems

on tribes will fill up the chapter A 1

Now we are going over to precise definitions and precise proofs

of basic theorems

A I I Ordering We define it as any correspondence (relation,

mapping, application) R, between elements of a manifold, satisfying the following conditions (6):

1) If x R y, Y R z, then x R z;

2) If x R y, y R x, then x = y and conversely;

3) R is not an empty correspondence, i e there exist elements

x, y with x R Y

For an ordering R we have:

if x EaR, then x R x

If R is an ordering, then a R = DR When dealing with an ordering,

it is convenient to write ::;;: instead of R

A.l.2 Expl

1) The correspondence ~, whose domain is the set of all real numbers, is an ordering The correspondence «) is not

2) The correspondence E ~ F, of inclusion of sets, whose domain

is e g the collection of all subsets of the euclidean plane, is an ordering

So is also the relation E ~ F

4 A General tribes (Boolean lattices)

3) The relation of equality with any domain, is an ordering 4) The correspondence between propositions p, q, defined as

"p implies q", is an ordering

S) The correspondence between positive integers a, b, defined by

"a, b have a common divisor greater than 1", is not an ordering

A.1.3 Lattice Let R be an ordering, which we shall write ~

Suppose that, given two elements a, b of G R, there exists an element c ,

a, b; join of a, b), and write a + b, (or a v b)

The operation will be termed ordering-addition

If for any two elements a, b the join a + b exists, we say that the ordering admits the sum, union, join of two elements

N ow suppose that given two elements a, b of G R, there exists

a third one d, such that

IX') d -::;;, a, d -;;;;, b,

(J') if d' -::;;, a, d' -::;;, b, then d' -::;;, d

In the case of existence, d is unique We call d product of a, b section 0/ a, b, meet of a and b) We shall write a b, a" b The operation

(inter-will be termed ordering-multiplication The product looks like an infimum

of two elements In the case of existence of the product of any two elements, we say that the ordering admits the product (intersection, meet), of two elements

two elements, the ordering will be termed lattice

We shall use the term soma (pI somata) to denote an element of

a lattice

A.I.4 Expl

6) The ordering 1) m [A.1.2.] is a lattice and we have

a v b =df max (a, b);

a" b =df min (a, b)

7) The ordering 2) in [A.1.2.] is a lattice where

E + F =df E v F (union of sets),

E F =df E " F (intersection of sets)

A General tribes (Boolean lattices) 5

8) Consider the set A of all closed rectangles on the plane with

sides parallel to the axes of a cartesian system of reference Let us define on A the ordering defined by the relation a ~ b of inclusion of sets Let us agree to consider the empty set as a rectangle Under these circumstances the ordering is a lattice

9) Let us make a similar agreement with the set of all closed circles

on the plane We shall not get any lattices

10) If in 8) we shall not consider the empty set as a rectangle, the corresponding ordering will not be a lattice

11) Consider the collection whose all elements are linear subspaces

of the euclidean three dimensional space This means that the elements are: the origin, the whole space, every straight line passing through the origin and every plane passing through the origin

Define the ordering as the inclusion of sets, confined to the above elements Under these conditions a + b will be the smallest linear space containing both a and b, and a b will be the intersection of the sets a, b (We notice that the join a + b is not the set-union of a and b)

The ordering is a lattice

A.1.4.1 Given a lattice R, let us consider a collection M of its

somata, which may be infinite It may happen that there exists a

soma b of R such that

IX) if a EM, then a ~ b;

(3) if b' EaR, and for all a E M we have a:S;: b', then b:S;: b'

Under these circumstances b is unique, and we say that the join

aEM

is meaningful (the union exists) and we define (1) as b It may also

happen that there exists a soma cEO R such that

IX') if a EM, then c:S;: a,

(3') if c' EaR, and for all a E M we have c' ~ a, then c' ~ c

Under these circumstances we say that the meet

a€M

is meaningful (the meet exists) and we define (2) as c If the set M is

given by an infinite sequence aJ, a2, , an, , we may write

respectively We do similarly if the collection M is finite

A.1.4.1 a We have defined infiniteoperations in a lattice (or even

in an ordering) A similar definition will be admitted for operations

on somata given by an indexed set {ai}, where the indices make up an

6 A General tribes (Boolean lattices)

abstract or defined non empty set] of some elements.1 We write

"smallest" soma in R, which means, an element 0, such that 0 < a

for all a EaR Such an element is unique, if it exists We call it zero

(or null-element), 01 R

It may also happen, that there exists the "greatest" soma, i e such soma, denoted by I, that a < I for all a EaR Such a soma

is unique whenever it exists We call it unit 01 the lattice R

A.1.4.3 Let R be a lattice admitting the zero and the unit Suppose that there exists a correspondence ~, one-to-one with domain and range a R, such that for all a EaR we have

A lattice may admit several correspondences like ~ (a)

A.1.4.3a Expl Take the example 1) in [A.1.4.] and define coa

as the whole space if a is the origin; the origin if a is the whole space; the line perpendicular to a, if a is a plane; the plane perpendicular:

1 From the logical point of view the indexed set {ail (i E J) is a function whose domain is ] and the range is a set of elements of a given ordering Every non empty collection of somata can be indexed by means of ordinal number: this by virtue of the axiom of Zermelo

A General tribes (Boolean lattices) 7

to a, if a is a straight line We get a complementary lattice Here the

complement is defined by means of orthogonality, but if we transform the space by means of a one-to-one linear correspondence, we shall change the orthogonality into something else, which however yields another way of defining the complementation in our lattice

A.1.4.4 We underscore the fact that the ordering-operations of

addition and multiplication, I a, II a depend not only on the elements

aEM aEM

of M, but on the totality of the lattice If we could increase or

diminish the domain of the lattice, the result of the operation can change

To be more clear, we should write

I(R) a, II(R) a, O(R), I(RI, co(R)a

If all sums and products exist, we call the lattice complete

A.1.5 We shall give a proof oftwo important laws for complementary

lattices, the so-called de Morgan laws

Theorem If (R) is a complementary lattice and M =1= g a collection

of somata of R, then the following are equivalent

Proof To facilitate the reasoning, consider M as an "indexed set"

{ai}, i E J, where] is a not empty set of some elements; see footnote p.6 We shall prove the first part of the theorem Put

supposing this exists By definition of the sum we have

(2) 1) ai-;:;;'b forall iEJ;

2) if ai :S b' for all i, we have

8 A General tribes (Boolean lattices)

From (2) we deduce [A.1.4.3.]

(4)

Suppose that

(4')

We have

cob < coai for all i E j

cob" < coai for all i

Let bill S coai for all i E j, and put b" =dfcob"' We have cob" = b"';

hence cob" < coai for all i E] Applying (5) we get b'" S cob Hence,

if b"';;:;; coai for all i, then b'll ~ co b This and (4) proves that

that the distributive law

(a + b) c = a c + b c

takes place for all somata a, b, c of R Then we call the lattice Boolean tribe (tribe, Boolean lattice The term generally used is Boolean algebra)

Thus the tribe is defined as a distributive complementary lattice

1 This proof is given by S NIKODYM

A General tribes (Boolean lattices) 9

A tribe is called trivial if it is composed of the single soma 0 = I

The most simple non trivial tribe is composed of two different somata

o and I

A.1.6.1 Expl.:

11) The ordering 1) in [A.1.2.] is not a tribe

12) The ordering 2) in [A.1.2.1 is a tribe Its zero is the empty set and its unit is the whole plane, coa is the set-complement

13) Let x, y, z be the axes of the cartesian system of coordinates

in the euclidean three-dimensional space Denote by (x, y) the plane passing through the axes x and y and define similarly the symbols

(y, z), (z, x) Let (x, y, z) denote the whole space 1.nd let 0 be the

set composed of the origin only We have eight elements: (x), (y), (z),

"(x, y), (y, z), (z, x), (x, y, z), O Define the ordering as in [A.1.4], 11), the zero as 0, the unit as tx, y, z) and coa as the ortho-complement

of a We have a tribe

14) We get a tribe by means of an analogous construction in the space with n dimensions (n = 1, 2, )

Remark Later we shall make a similar construction in the ordinary

Hilbert-space with infinite dimensions

Expl 15) The· expl 8) in [A.1.4.] with set-complement is nota tribe

A.2 Boolean algebra There exists a vast theory of orderings,

lattices and tribes, but we shall confine ourselves to quote, without proofs, several laws governing the tribes and we ask the reader, interested

in details, to consult special monographs and papers (6), (8), (9), (12), (13) These laws are the following:

10 A General tribes (Boolean lattices)

A.2.3a De£ We define the s~tbtraetion ot somata as follows:

A.2.4 Def Two somata a, b are called disjoint whenever a b = O

A2.5 The following are equivalent:

I) a < b, II) a· b = a, III) a + b = b

A.2.6 If a:S:; b, e < d, then a e < b d and a + e ;2; b + d;

absa, asa+b, a~a;

if b < a, then a = b + (a - b)

The proofs of the statements [A.2.3b.], [A.2.5.], [A.2.6.] are straight forward

A.2.6.1 We shall need some more complicated formulas, which

we shall provide with proofs

Theorem If

then

a - e = (a - b) + (b - c)

Proof.! We have [A.1.4.3.],

By [A.2.3 b.] a = a b + (a - b); hence multiplying both sides by coe, we get a· coe = [(a - b) + a b] coe and by [A.1.6.]:

a coe = (a - b) coe + a b· coe and, by [A.2.3a.],

(2) a co e = a co b co e + a b co e

As by (1) we have: cob:S coe, and by hypothesis we have b sa,

therefore we have by [A.2.5.] and by (2):

A General tribes (Boolean lattices)

A.2.6.2 Theorem If

then

al - a" = (al - a2) + (a2 - aa) + + (a,,-l - an)

The terms on the right are disjoint

Proof We rely on [A.2.6.1.] and apply induction

A.2.6.3 Theorem

I.-(1) I an = al + [a2 - al] + [aa - [aa - (al + a2)] + +

n-l

+ [a" - (al + a2 + + ak-l)],

for k = 1, 2, The terms on the right are disjoint

Proof Suppose that for a given k we have (1) We have

Proof Put b" =df al + a2 + + al.- \Ve have a,,:::;: bk, hence

cob,,:::;: coa" As bl.-+ ahl:::; 1 = bk + cobk :::;: b" + coak, we have

bl.-+ ahl ;;:;; (b" + ak+l) (bk + coa,,)

Hence

(1 )

and we have

= b" bl.-+ ah·+l· b" + b k · coa" + a"+l· co a"

= bl.-+ a"+l coa" + bdahl + coal.-)

= bl.- + al.-+l coak = bl.-+ (al.-+l - ah·)

al + a2 = al + (a2 - all·

Now, suppose that for a given n:

(2) bIt = al + a2 + + all:::;: al + (a2 - al) + + (an - an-I)

12 A General tribes (Boolean lattices)

We get from (1)

bn + an+I ~ bn + (a,,+1 - all);

hence, from (2),

b" + an+I < al + (a2 - aI) + + (an - an-I) + (a'HI - all),

i.e al + + all +I Sal + (a2 - aI) + + (an+l - an)

The theorem is established, because the converse inequality is clear A.2.6.5 Theorem If for somata an, bll of a tribe, (n = 1,2, ),

By the definition of sums we have for all n:

In a similar way we prove that a < b, which completes the proof

A.2.7 Remark We see that the above rules of the Boolean algebra are very similar to those which take place in the theory of general sets But the Boolean algebra is not interested in points of the sets, even if such points are available It may be called theory of "sets without points"

A.2.8 Remark One can give the foundation of the theory of tribes by starting with a well selected collection of formal rules [A.2.1.]

A General tribes (Boolean lattices)

to [A.2.3.] and by defining the ordering by [A.2.S.]:

a s b· =dr' a· b = a, or by a ~ b· =dr' a + b = b

A.2.9 Remark If, given a tribe and its algebra, we replace the addition a + b by another one a -+- b, defined by

a -+- b =d,(a - b) + (b - a),

and by keeping the multiplication, the tribe will be reorganized into

a commutative ring with unit We shall call it Ston,e's ring (9)

If we order T by means of the relation of inclusion (~) of sets, we get

a tribe which is not denumerably additive

Indeed, there does not exist the somatic union of the half-open intervals

2P+ 1 ' zp/' p=1,2,

14 A General tribes (Boolean lattices)

But some denumerable somatic unions may exist; for instance

(n = 1, 2, )

This join equals (0,1)

A.3.2 Def It may happen that the tribe admits all finite and infinite joins Such a tribe is called completely additive

A.3.3 Expl The tribe of all subsets of (0; 1) with inclusion of

sets as ordering relation

A.3.4 Theorem In a tribe the following are equivalent:

I) All denumerable sums exist;

II) all denumerable products exist

If we take account of DE MORGAN laws [A.1.5.], we get the existence

of tI an too Thus I) implies II) A similar proof is for the implication

(This is a kind of associative law.)

Proof It is not difficult to prove the assertion:

A General tribes (Boolean lattices)

this for all m = 1, 2,

Applying the definition of the sum, we get

A.3.6 Theorem If (T) is a tribe denumerably additive, then

the "infinite" distributive law takes place:

16 A General tribes (Boolean lattices)

By definition of the lattice-sum, we have

an;;:;; b for all n

Hence, taking any k ;:s 1, we have

A General tribes (Boolean lattices)

Now let b k < b' for all k We have

Hence, by (3), we have e:S;;: e'

Since, by (4), ak < c for all k, and since we have the implication:

00

"if ak:S;;: c' for all k, then e:S;;: e'," it follows that 1: ak exists and equals c The theorem is established k-l

A.3.S Theorem If the tribe (T) is denumerably additive, then

the following distributive law takes place for any b, ai, a2, , an, E T:

(li an) + b =n~(an + b)

The proof'is based on the theorem [A,J.6.], and on the infinite de Morgan laws [A H.]

A.3.9 Theorems similar to [A,J.6.] and [A.3.8.], but involving non-denumerable addition and multiplication can be proved in an

18 A General tribes (Boolean lattices)

anologous way, so that these more general theorems can be considered corollaries to the proofs of [A.;3.6.] and [A.3.8.] Especially the theorems are useful for completely additive tribes [A.3.2.J

A.4 The role of equality (12) Every mathematical theory possesses a specific notion of equality of its elements, denoted usually

by the sign (=) and having the formal properties: reflexiveness, symmetry and transitivity This notion, according to the case, may

be introduced axiomatically, or defined in a suitable way The operations performed on elements and the relations between them are invariant with respect to this equality For instance, in the theory of tribes, we

For example, if we have a ~ b, we can replace a and b by a' and b'

respectively, whenever a = a' and b = b' We call (=) the governing equality in the given theory

A.4.l If E is a set of elements, and a E E, a = a', then a' E E

The relations have an analogous property If we have two kinds of elements a, b, and A, B for which the governing equalities are

(~) and (~) respectively, and if S is a correspondence between the elements of the first kind a, b, and the elements A, B, of the second kind, we should have:

"if a S A and a ~ a', A ~ A', then a' SA'."

We say that S is (~)-invariant in its domain and (~)-invariant

in its range

To be very strict we should say "(=)-set", instead of "set" We

also should say (~), (~)-correspondence, instead of "correspondence"

A.4.2 The governing equality induces the notion of the unique element x satisfying the condition W ( x ), (1) The statement:

"there exists a unique element x satisfying the condition W(· x.)"

means:

"there exists an element x for which W ( x.) holds true, and if we

have W(· x'.) and W( x" ), then x' = x"

Similarly the statement:

"the unique element x, satisfying the condition W ( x ), has the property q:>('x,)"

means:

"there exists an element x, such that W ( x ) and q:> ( x ); and if W ( x' )

and It' ( x" ), then x' = x"."

A General tribes (Boolean lattices) 19

We should say "(=)-unique element" instead of "unique element" In addition to above the notion of equality conditions the cardinal number of a given set of elements We should say "(=)-cardinal of a set", instead of "cardinal number of the set" A.4.2a Expl The difference between the geometrical free vectors, sliding vectors and fixed vectors is conditioned by various kinds of equality If we deal with free vectors, there is an infinity of geometrical vectors which are equal to one another, but the set of all vectors geo-metrically equal to a given vector ii, has the cardinal number = 1

If the vectors were "fixed vectors", the cardinal number of the above collection would have the power of continuum

A.S In many simple cases the above remarks are not very important, but the situation is different, if we need to introduce many kinds of equalities

Usually we have in a theory a basic notion of equality, and we call other derived kinds of equality: equivalences

A.S.1 Till now we did not pay attention to the logical difference between an ordering R and its domain

DeL Now, since we shall have some more subtle reasonings, we shall agree to denote the ordering by (R) and its domain by R Really these two notions are logi~ally differe11t, smce R is a set and (R) a

relation

A.6 Homomorphism of tribes (12), (13) Let (T), (T') be two

tribes, and let

xA x'

be a correspondence which carries the domain a T onto a subset of

o T' in such a way that the operations and relations for somata of (T) go over to the corresponding operations and relations for somata

of (T') For instance: if x+y=z in (T), xAx', yAy', then x' + y' =' z' in (T'); if x < y in (T), then x' ~' y' in (T') [The notions

considered in (T') are provided with primes.] In these circumstances

we say that A is a homomorphism from (T) into (T')

If A carries the domain 0 (T) into the whole 0 (T') we say

"homomorphism from (T) onto (T')" The homomorphism should be

invariant in 0 A with respect to the equality (=) governing in (T),

and A should be (=')-invariant in D A

If the correspondence is 1 -r 1, we call A isomorphism To be very

strict we should say "(=), (=')-homomorphism" and even tions and order - (=), (=')-homomorphism from (T) into

"opera-(or onto) (T'l"

A 7 Genuine subtribes and supertribes Let C be a collection

of some elements with a relation of equality(=C) of elements, satisfying

20 A General tribes (Boolean lattices)

the usual conditions of identity The subsets of C will be also termed collections and we shall speak of elements of C The collection C will

be a kind of substratum to what follows

Now suppose we have two tribes (T') , (T") such that T', T" are

subcollections of C, so we can write T' ~ C, T" ~ C, and such that

T' ~ T" These subcollections are invariant with respect to the equality (=C); i e if a E T', a =C b, then bET' and if IX E T", IX =C {J, then {J E T"

Having that background, suppose that in the tribe (T') we have the governing equality (='), and in (T") the governing equality (=") , which may differ from (=C) We suppose that

if a,bET', a=cb, then a='b,

and if IX, {J E T", IX =C {J, then IX =" {J, (but not necessarily conversely) Under these circumstances, we define the following notion, which will be important in the sequel, (13), (14)

We say that (T') is a finitely genuine subtribe of (T") or (T") is a finitely genuine supertribe of (T') whenever the following conditions

are satisfied for the somata a, b, c, , 0', I' of (T'):

10) a +' b =' c is equivalent to a +" b =" c;

2°) a.' b =' c is equivalent to a " b =" c;

3°) I' =" I";

4°) 0' =" a"

(Single primes refer to (T') and double primes refer to (T")

A.7.1 One can prove that the above four conditions are dent, (13), and that they imply the following rules for somata of (T'):

indepen-a =' co'b is equivalent to a="co"b,

a -' b =' c is equivalent to a -" b =" c,

a +' b =' c is equivalent to a +" b =" c,

a -::;" b is equivalent to a <" b

A.7.2 Def If in addition to 1°), 2°),3°) we have not only T' ~ T"

but T'~" T", we say that (T') is a finitely genuine strict subtribe of (T") and (T") is a finitely genuine strict supertribe of (T')

A.7.3 Expl Take as substratum C the collection of all subsets

of the interval (0, 1> =d,{X 10< x < 1}, a"nd denote by (=C) the

equality of these sets, (identity of sets, restricted to C) Now consider

all finite unions of sets {xja < x <b}, (0 ~ a, b ~ 1) If we order these 'collection by the relation of inclusion of sets, we get a tribe (T'),

whose governing equality (=') is just (=C); restricted to T' Let us

call the somata of (T) figures

A General tribes (Boolean lattices) 21

On the other hand consider all Lebesgue-measurable subsets of (0, i) where the ordering is defined by

E S" F· =df· meas(E - F) = o

The governing equality will be

E =" F· =d{· meas[(E - F) + (F - E)] = O

Thus obtained ordering is also a tribe, which wiII be denoted by (T")

We shaH see that (T') is a finitely genuine subtribe of (T")

To prove thai we shall show that the conditions 10), 20), 30), 4°)

in [A.7.] are satisfied Consider the col1ection C and its elements, which

are subsets of (0, 1) E =" F means that E, F do not differ but

by a set of measure O This is equivalent to the equality

EvP=Fvq

where p, q are some sets of measure O

It follows that if for figures f, g, h we have f v g =' h we have also f v g =" h If we have f n g =' h, then f n g =" h The zero 0" of (T") is represented by any set of measure 0, and the unit I"

is represented by any set (0, i) - P where meas p = O The zero

of (T') is {} and its unit is (0, i)

On the other hand, if we have for figures p, q, r, P +" q =" r, then

p + q = r, i e P +' q =' r, follows, because the only figure with

measure 0 is the empty set of points Similarly, if p " q =" r, we must have p n q = r, hence p.' q =' r

Thus we have proved that (T') is a finitely genuine subtribe of

(T") We do not have T'~" T" Indeed this relation requires that T' be invariant with respect to the equality '( =") This however is

not true, because a set of measure 0, different from 0, does not belong

to T' We can change (T') into a tribe (T~) which would be a strict

subtribe of (T") It suffices to consider all measurable sets E which

do not differ from figures but by a set of measure O It is, we replace every figure f by the collection of all sets f - p v q, where p, q are sets

of measure O

Then we shall have T~ ~" T", so (T~) is a strictly genuine subtribe

of (T") The tribe (T~) is isomorphic to (T')

A 7.4 Remark If (T') is a finitely genuine subtribe of (T") ,

but not strictly genuine, we can by a simple modification obtain a

22 A General tribes (Boolean lattices)

tribe (Ti) which is isomorphic to (T') and at the same time a finitely strict genuine subtribe of (Til)

Indeed, let 58 be a correspondence which attaches to somata a

of (T') all somata IX which are (=")-equal to a, and belong to (Til)

Then 58 can be considered as the (='), (=")-isomorphism a 58 IX, The tribe 58 (T') =dr Ti is the desired tribe

A.7.5 Def If in [A.7.] we replace the conditions 1°, 2° by the following

A.7.6 Remark In this work we shall have the need of performing,

in a reasonable way, denumerably infinite operations on a finitely additive tribe (T') Now this can be done if (T') is a finitely genuine

00

subtribe of a denumerably additive tribe (Til) We just use ~" 58 (all)

n-l

and we shall say that we took infinite operations from (Til)

A.9 Ideals in a tribe (12), (13) We know [A.2.9.], [A.2.1O.],

that a tribe (T) can be reorganized into a commutative ring with

unit (Stone's ring) Consequently we can consider ideals in that ring,

as we do in any ring in modern algebra

De£ Therefore we shall call ideal in (T) any non empty subset J

of T satisfying the following conditions:

1) if a, bE J, then a -' b E J (a -' b means algebraic subtraction

We see that the set T is an ideal in (T) and also the set, composed

of the single soma 0, is an ideal In general we have 0 E J

A General tribes (Boolean lattices) 23

A.9.2 Def An ideal j in (T) is called denumerably additive whenever

the following happens:

These ideals are important in the case where the tribe (T) is

denumer-ably additive An ordinary ideal is called sometimes finitely additive,

to stress that it does not need to be denumerably additive

of (T), defined by

a -;;;;, J b = df • a - b E j,

and also a special equality (=J) called equality modulo j, a =J b,

defined by a ~ J b, b < J a The relation a =J b is equivalent to

a + b = (a - b) + (b - a) E j

A.9.3a Theorem The relation a -;;;;, b implies a <J b Indeed

a - b = 0 E j It follows that the relation a == b implies a = J b

A.9.4 To prove properties of this new equality we mention the following formulas, whose proof can be easily found by drawing two overlapping circles:

24 A General tribes (Boolean lattices)

A.9.S Theorem The correspondence «J) is an ordering, and (=J)

obeys the laws of identity

A.IO Theorem The ordering (~J) with domain T is a tribe

Proof First we shall prove that the ordering admits joins

Let a, bET I say that a + b is the (~J)-union of a and b, i.e

To prove that a + b ::;: J c, it suffices to prove that (a + b) - c E ],

[A.9-3] But this follows from

and it follows from (1), [A.9.1.]

A.IOa This being done, we shall show that the ordering (:SJ)

admits meets We shall prove that a· b is the meet a.J b We have

A General tribes (Boolean lattices)

hence to

(c - a) + (c - b) E ]

This however is true because of (2)

Thus we have proved that

i e (:;:;; J) is a lattice

a +J b =J a + b, a.Jb=Ja.b,

A.IOb The lattice has the zero 0 and the unit I

We shall rely on the following remark:

In a distributive lattice with 0 and I there cannot exist two different notions of complement

We have

a + co a = I, a co a = O

Hence relying on [A.9.3.], [A.1Ob.] and [A.10a], we/get

a+Jcoa.=JIJ=I,1 a.J coa· =J OJ = O

We shall prove that if a:;:;;J b, then cob sJ coa We have, by definition

26 A General tribes (Boolean lattices)

which gives, [A.9.3J

cob "'2,.J coa

Since the soma coa abeys all laws needed for the (=J)-complement,

and since there are no two different notions of complementl , in a distributive lattice with zero and unit, therefore "co" is the complement

in the lattice (T)J Thus, we have proved that «J) is a tribe We shall

denote it by (T J)

A.IO.1 Theorem We have also proved that

a+Jb=Ja+b, a.Jb=Ja·b,

IJ =J I, OJ =J 0, coJa =J coa

Thus in performing operations in the new tribe (T J ), we can write

these operations as if they were in (T), only the sign (=) must

be changed into (=J) and (~) into ("'2,.J)

A.IO.2 Theorem The operations of addition, multiplication and complementation in the ordering «J), i.e in (TJ) are (=J)-invariant The following proof may be useful

Proof We start with addition Let a +J b =J C and let a =J aI,

b =J bl , C =J Cl We shall prove that

A General tribes (Boolean lattices)

We have for some somata p, q of ], [A.9.4a],

28 A General tribes (Boolean lattices)

A.I0.2b We are going to prove the (=J)-invariance of the operation of complementation Let

b =J coJa, a =J aI, b =J bl ;

we shall prove that

To do that, apply [A.9.4a] We get

which gives, [A.9.4a V)],

Applying [A.10.1.] we get

A.I0.3 Theorem We also have the following The statement

1 a <J b is equivalent to the statement;

II there exists p E J such that

a + p :s:;: b + p hence II follows

Let II., i.e let p E J with a + p:s:;: b + p

A.tO.4 Theorem The relation a 5:,J b is (=.1)-invariant

Proof Let a 5:,.1 b and a =.1 aI, b =.1 bl ; a -;;;;'.1 b is equivalent to

a +.1 b =.1 b,

because that statement is equivalent, [A.10.2.], to

al +.1 bi =.1 bl

This however is equivalent to

A.ll To perform (-;;;; .I)-operations we use simply the somata of

(T) and operations on (T) but changing the signs (=), (-;;;;.) into (=.1)

and (-;;;;'.1) respectively In any formula we are allowed to replace somata

a, b, by any others aI, bl , which are (=J)-equal to a, b,

The tribe (5:,.1) will be termed tribe (T) taken modulo J and it has already been denoted by (T J )

A.tt.t If the ideal I is composed of the only soma 0, the tribe

(T.1) will coincide with (T), and if I = T, then the ordering (T J) will

be trivial, because all somata of (T).1 will be (=.1)-equal, and equal

to O

A.t2 Given an ideal I in (T), we have defined the tribe (T) module I

by a change of ordering, and consequently by changing the equality

(=)-governing in (T) into another kind of equality, viz (=.1) There

is however an other way of using the ideal I in defining a new tribe, viz by forming so-called equivalence-classes We are going to give a sketch of the corresponding theory Given a soma a of (T),

let us consider the set

A =d,{x[xET, x-+-aEI}

We call A the I-equivalence class corresponding to a, and denote it by

[a] The following are logically equivalent:

I) x E A, II) x -+- a E I

30 .-\ General tribes (Boolean lattices)

A.12.1 Theorem If A is the equivalence class corresponding

to a, a + bE], then A is also the equivalence class corresponding

to b The equivalence classes corresponding to somata of (T) are either

disjoint or identical The somata of [a] are called representatives ot [a]

A.12.2.: Def We define for equivalence classes the operations of addition and multiplication, in the following way If a is a repre-

sentative of the equivalence class A and ba representative of the valence class B, we consider the equivalence class C corresponding to

equi-a + b, and call C the J-join of A and B:

[a + bJ = rid x + y I x E A , y E B}

One can prove that C does not depend on the choice of representatives

[a + bJ =rif [a] + [bJ

In a similar way we define the meet A ·B of the equivalence classes, and

for their complementation We prove that just defined operations

constitute a Boolean algebra, so we can define the corresponding ordering (:sJ) on the set of 3.11 equivalence classes We obtain a new tribe, called quotient tribe (T) I]

A.12.3 Theorem One can prove that the tribes (T)J and (T)IJ

are order and operations isomorphic

We shall confine ourselves to the above sketch, leaving the details

to the care of the reader

A.13 Till now we have considered general ideals in a tribe Now

we are going to study the situation where we have a denumerably

additive ideal] in a denumerably additive tribe (T), [A.9.2.J

Theorem First of all we shall prove that in this case the tribe (T)J

i.e (T) modulo] is also denumerably additive, i.e for any somata

A General tribes (Boolean lattices) 31 Proof Put a =df ~ an We have an ~ a for every n = 1, 2,

The remaining part of the thesis follows

A.13.1 Theorem If (T) is a denumerably additive tribe and ]

A.13.2 If (T) is a denumerably additive tribe and] a denumerably

additive ideal in it, then the denumerable joins are (=J)-invariant Proof Suppose that an =J b n , (n = 1, 2, )

By [A.9.4a.] there exist Pn E] such that