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

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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 1st edition 1966
Library of Congress Catalog Card Number 66·27977

Tide No.5 I 12

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Dedicated to my wife
Dr. Stanislawa Nikodym

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Preface
The purpose of this book is to ,give the theoretical physicist a
geometrical, visual and precise mathematical apparatus which would
be better adapted to some of their arguments, than the existing and
generally applied methods. The theories, presented in this book, are
based on the theory of Boolean lattices, whose elements are closed
subspaces in the separable and complete Hilbert-Hermite-space.
The first paper, in which the outlines of the said mathematical
apparatus is sketched, is that of the author: "Un nouvel appareil
mathematique pour la theorie des quanta."]
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
The paper constitutes the content of four lectures by
and 13, (1947) at the Institut HENRI POINCARE in

2 Bd.37, 1951, P.278/279.
3 Berlin/Gottingen/Heidelberg: Springer-Verlag.
matischen Wissenschaften 52. (1954), XII
460 pp.

+

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XI, fasc. II, pages 49-112.
the author: February 4,6, 11
Paris.
Die Grundlehren der Mathe(see the footnote p. 75).


VIII

Preface

later been simplified, generalized and applied in several papers by
the author. The present book can be considered as a systematic synthesis 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 Hilbertspace. 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 geometrical 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 spectraltheorem included,
2) with the theory of Lebesgue's measure and integration,

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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
natural numbers.
The book contains 29 chapters, which are only partly depending on
one another. They are labelled with letters:
A, A I, B, B I, C, C I, D, D I, E, F, G, H,

J, J I, K, L, M, N, P, PI, Q, Q I, R, R I,
S, T, U, W, W 1.
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.

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x

Preface

But my most hearty thanks I am owing to my wife Dr. STANISl.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.
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,
July 10. 1966


Dr. 0TTON MARTIN NIKODYM

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List of chapters
A.
A 1.
B.
B 1.
C.
C 1.
D.
D 1.
E.
F.
G.
H.

General tribes (Boolean lattices)
Special theorems on Boolean lattices
Important auxiliaries. . .
General theory of traces .
The tribe of figures on the plane
The trace-theorem . . . .
The lattice of subspaces of the Hilbert-Hermite space.
Tribes of spaces . . . . .
Double scale of spaces . .
Linear operators permutable with a projector.
Some double STIELTJES' and RADOX'S integrals.

Maximal normal operator and its canonical representation .

J.
J 1.

Operators N j(x) =df q; (x) . j(i) for ordinary functions! . .
Operational calculus on general maximal normal operators.
Theorems on normal operators and on related canonical mapping
K.
L.
Some classical theorems on normal and selfadjoint operators.
M.
Multiplicity of spectrum of maximal normal operators
N.
Some items of operational calculus with application to the resolvent and spectrum of normal operators
P.
Tribe of repartition of functions
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 . . . . .
R.
R1. Summation of quasi-vectors in the separable and complete
Hilbert-Hermite-space
. . . . . . . . . . . . . . .
General orthogonal system of coordinates in the separable and
S.
complete Hilbert-Hermite-space . . . . . .


T.

U.

DIRAC'S Delta-function . . . . . . . . . .
Auxiliaries for a deeper study of summation of scalar fields

44
130
145
173
206
252
285
335
386
391
402
428
450
462
502
521
560
586
622
644
675
689
708

718
724
760

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
admissibility. Square summability of fields of numbers

884

References . . .

942

Alphabetical index

947

W.

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805


Chapter A

General tribes (Boolean lattices)
A.t. This first chapter contains the fundamentals of the theory

of Boolean lattices.
We start with intuitive remarks; the precise setting will
follow later at [A.1.1.J.
The Boolean lattices constitute an important item in modern
functional analysis, though they did not occur in textbooks of Quantum
Mechanics as an essential tool. Nevertheless special Boolean lattices,
whose elements are invariant closed subspaces of the Hilbert-space
play a quite important role and give a purely geometrical basis to
selfadjoint Hermitean operators. The said importance has been shown
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 complementation, obeying just the same formal rules, as in the theory of sets,
but with the exception that the relation of belonging of an element
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 (propositional function) (1), involving two variables, similarly as the set is a
notion attached to a condition (propositio~al function), with one variable
only. Thus e.g. as the condition for numbers 3x - ~
x


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=

5 with the


2

A. General tribes (Boolean lattices)

variable x generates the set {x 13 x - : = 5}, so the condition 3x - 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 condition 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 1v (. 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.

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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 reconstructs 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.

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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 GR, there exists an element c ,
such that
IX) a ~ c,
b -::;;, c,
(J) if a ~ c', b ~ c', then c -::;;, c'.
In the case of existence, c is unique. The element c is something
like the supremum of a, b. We shall call c sum of a and b, (union of
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 (intersection 0/ a, b, meet of a and b). We shall write a . b, a" b. The operation
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.
Lattice. DeL If an ordering admits sums and products of any
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
E .F

=df

F

E" F

(union of sets),
(intersection of sets).

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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

(1)

~a

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

IJa

(2)

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
00

~an'

n-1

00

nan

,,-1

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


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6

A. General tribes (Boolean lattices)

abstract or defined non empty set] of some elements. 1 We write
I ai, II ai respectively.
iEJ

iEJ

iC.1.4.1 b. The lattice-addition and multiplication obey the following laws, (8):
x· y = y. x;

x . (y . z) = (x . y) . z;

x . (x

x

+ y) = x;

+ (y + z) =
x

(x


+ x .y =

+ y) +z;

x.

In addition to that the following are equivalent:

(4)

I) x S y;

II) Y = x

+ y;

III) x· Y = x.

Conversely, if we have an algebra with operations of addition and
multiplication obeying the laws (3), then if we define the ordering
«) by means of II) or III), we get a lattice whose operations are just
those which are defined in [A.1.3.]. Thus we can reorganize a lattice
into an algebra, and conversely, the algebra into a lattice.
A.1.4.2. Let R be a lattice. It may happen that there exists the
"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
IX) a + ~ (a) = I,
(J) a· ~ (a) = 0,
y) if a < b, then ~(b) S ~(a).
If such a cQrrespondences ~ exist and if we have chosen one of
them, we shall call the lattice complementary, and we call ~ (a) complement
01 a.

Instead of
a

~(a)

+ co a = I;

we agree to write coa. Thus we have
a . co a

=

0;

co(coa)

if a S b,

=


then co b S co a;

a.

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.

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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.
aEM

aEM

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
I) I a exists,
aEM

II) II coa exists.
aEM

In the case of existence we have
co I

aEM

a = II coa.
aEM


We also have the following equivalence of statements:
I') II a exists,

II')

aEM

I

aEM

coa exists.

In the case of existence we h'we
co II a

1; coa.

=

aEM

aEM

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
(1)

b=drIa=Iai,

aEM

.

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

(3 )

b ~ b'.

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8

A. General tribes (Boolean lattices)

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

(4)


cob

<

coai

for all i E . j.

Suppose that

(4 ' )

cob"

<

coai

ai

<

b"

We have
hence, by virtue of (3),

for all i.
for all


$;

b ;; b".

Consequently
(4")

s

cob"

cob.

Thus we have proved that if
co b" ;S; co ai

for all

$,

then we have

(5)

cob"

<

cob.


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
cob = n coai. Hence n coai, by [A.1.4.1.], exists and is equal to cob.
iEJ

By virtue of (1) we get
ncoai=coIai,
i

(iEj),

.

which constitutes the thesis of the first part of the theorem.! Now
suppose that n coai exists. Put
cob =df n coai,
i

and perform the reasoning similar to the above one. We get the
existence of I ai and then I).
i

The second part of the theorem is a consequence of the first. It
suffices to replace a by co a, and conversely.
A.1.6. Tribes. Let R be a complementary lattice and assume
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.

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9

A. General tribes (Boolean lattices)

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:
A.2.1.
a

+a =

a;

a+O=a;
a·a

=


a;

a·O =0;

a

+ b = b + a;

a

a+I=I.
a . b = b . a;
a·I

=

a . (b . c)

=

(a . b) . c;

a.

A.2.1a.
(a

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


+ b) . c = a c + be,

a.b

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

A.2.2. De Morgan laws:
co (a

+ b) =

coa . cob,

co (a . b) = coa

A.2.3.
col = 0,

coO

=

I,

+ cob.
co(coa)

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=


a.


A. General tribes (Boolean lattices)

10

A.2.3a. De£. We define the

s~tbtraetion

ot somata as follows:

a-b·=d(·a.cob.

A.2.3b. The following laws are valid:
a-b=a-a·b,
a
a

+b=

(a - b)

a-a=O,

+ a . b + (b -

+ b = a + (b - a) ,

a = a . b + (a - b).

a),

A.2.4. Def. Two somata a, b are called disjoint whenever a . b = O.
A2.5. The following are equivalent:

II) a· b = a,

I) a < b,

III) a

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

if b < a,

asa+b,

then

a

+ b = b.
a + e ;2; b + d;

a~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.],

(1)

coa

+

< cob S coe.

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.],
a . co e


(2)

+

= 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 . co e = a . co b + b . co e .
Hence by [A.2.3 a.] we get
a- e
1

=

(a - b)

The given proof is by S.

+ (b -

c)

NIKODYM.

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q.e.d.



A. General tribes (Boolean lattices)

11

A.2.6.2. Theorem. If

(n=2,3, ... ),

al~a2~···~all'

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.-


n-l

an

=

+ [a2 + [a" -

al

+ [aa - [aa - (al + a2)] + ... +
(al + a2 + ... + ak-l)],

al]

for k = 1, 2, ... The terms on the right are disjoint.
Proof. Suppose that for a given k we have (1). We have

I

(2)

I.-

n=l

all

+ [ak+1-


+ a2 + ... + ail')]

(al

= b

+ (ahl- b),

where
b =df I

I.-

n=l

an·

Applying [A.2.} h.], we see that (2) equals
"+1

I.-

b

+ a"+l = I=1 all + a"+l = I=1 an,
Il

11


which completes the proof.

A.2.6.4. Theorem.

+ (aa - a2) + ... + (an - an_I).
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,,)
= b". bl.- + ah·+l· b" + bk · coa" + a"+l· co a"
= bl.- + a"+l . coa" + bdahl + coal.-)
= bl.- + al.-+l . coak = bl.- + (al.-+l - ah·).
I

It

"_1

ak = al

+ (a2 -

Hence
(1 )
and we have

al

all

+ a2 =


al

+ (a2 -

all·

Now, suppose that for a given n:
(2)

bIt = al

+ a2 + ... + all:::;: al + (a2 -

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al)

+ ... + (an -

an-I).


12

A. General tribes (Boolean lattices)

We get from (1)

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


hence, from (2),

+ an+I < al + (a2 - aI) + ... + (an - an-I) + (a'HI - all),
i.e. al + ... + a + I Sal + (a 2 - aI) + ... + (an+l - an).

b"

ll

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, ... ),
we have
k

k

1: an = 1: bn

(1 )

n-l

for all k

n-l

=

1, 2, ... ,


then, in the case of existence of the denumerable sums
00

00

1: an, 1: bn,
n=l

n=l

we have

00

1: an = 1: bn.

n=l

11,=1

A.2.6.6. Proof. Suppose the existence takes place. Put
00

a

=df

1: a,,,


n-l
By the definition of sums we have for all n:

hence
k

1: an <

By (1) we have

n-l

a for every k

k

1: bn S

n-l

a;

=

1 , 2, ...

bk S a for every k;

hence
00


hence, by definition of the sum

1: bk ,

k-l

we get

bs a.

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.]

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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,

a ~ b· =dr' a

or by

+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).
We have:
a -+- a

=

0;


a

+ b = a -+- a . b -+- b;

co a

=

I -+- a;

the equation
a-+-x=b

has the unique solution a-+- b, so for the corresponding subtraction
(--'-) we have b--'-a = b -+- a.
We shall call the operation' a -+- b algebraic addition in discrimination
with a + b, which we shall call somatic addition. The most used ferm
is symmetric difference.
A.2.tO. Remark. If we have such a ring, we can go back to
the Boolean algebra, by defining the somatic addition by
a

+ b· =df' a -+- a b -+- b.

A.2.t1. Def. A soma a of a tribe is called atom, whenever the
following conditions are satisfied:
1) a =1= 0;
2) jf b :s;; a, then either b = 0 or b = a.
Expl. In the example 2) in [A.1.2.] the ordering is a tribe. Its atoms

are sets composed of single points.
A.3. Def. There are tribes which admit all denumerable joins of
their somata. Such tribes will be termed denumerably additive.
A.3.t. Remark. There are tribes which are not denumerably
additive.
Expl. Let T be the class of all finite unions of the sets (a, b) of real
numbers, where 0 ~ a ~ 1, 0 s b s 1,
(a, b) =d'{xla < x

< b}.

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
(

1

1 \

2P+ 1 ' zp/'

p=1,2, ...

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A. General tribes (Boolean lattices)


14

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.
00

Proof. Suppose that all sums
all sums

~

~

n-1

coa n .

n=l


If we take account of

of

tI an too.

DE MORGAN

an exist; then there also exist

laws [A.1.5.], we get the existence

Thus I) implies II). A similar proof is for the implication

ll-l

II

1.
A.3.5. Theorem. Let R be a lattice, where all denumerable sums
exist.
Then
->-

00

2: (an + bll)

00


00

2: an + 2: b

=

1&-1

1&-1

ll •

11=1

(This is a kind of associative law.)
Proof. It is not difficult to prove the assertion:
00

(0)

00

~ an :::::: ~ (an

n= 1

It =

=


Indeed, we have for all k

1

+ bn).

1, 2, ...
00

ak

+ bk :::::: ~ (an + b

ll ) .

n=l

Since ak < ak

+ bk , we

get for all k
ak :::::: ~ (an

hence, by definition of sum,
00

1l=1


+ bTl);

00

~ an ~ ~ (an

n=l

H=l

+b

ll ) ,

so the assertion (0) is proved.

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15

A. General tribes (Boolean lattices)

Similarly we have
00

(0.1 )

~ b" ~ ~ (an


11.=1

n=l

+ bn) .

Adding (0) and (0.1) sidewise, we get
00

(1 )

00

~ an

ll=l

00

+ ~ bn:S: ~(an + bn).
n=1

11.=1

On the other hand we have
00

00

am:S: ~ an,

n-l

hence, [A.2.6.J,

bm ~ ~ bn ;
n ... l

00

am

00

+ bm < n-l
~ an + ~ bn,
n~1

this for all m = 1, 2, ...
Applying the definition of the sum, we get
00

00

(2)

~(am

00

+ bm):S: n-l

~ an + ~ bn·
n-l

m-l
From (1) and (2) the theorem follows.
A.3.6. Theorem. If (T) is a tribe denumerably additive, then
the "infinite" distributive law takes place:
00

00

b.~an=~(b.an)

Proof. We have
00

b· I

n-l

n-l

n-l

for all an,b.

00

00


an

= b· I an· I = (by [A.1.4.3.)]
n=l

=

b· I an· (b
11.=1

+ cob)

b . I (an· b + an . cob).
00

=

n=l

because the finite distributive law is valid in a tribe [A.1.6.J.
Applying [A.3.5.J, we get
b ·n~ an

=

b· [J'I(a n b)

+,~(anCOb)];

hence, by the finite distributive law,

00

00

00

b . ~ an = b . ~ (an b) + b ~ (an· cob).
n-l
n-l
n-1
Now an· cob:S: cob for all n, [A.2.6.J, hence, by virtue of the definition
of infinite sum:
•

(1 )

00

~

n-l

(an· cob):S: cob.

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16

A. General tribes (Boolean lattices)


Hence
00

b·

From (1) it follows

~
n-l

(an' cob) < b . cob
00

=

O.

00

b . ~ a" = b . ~ (an b) .
n-l

n-l

00

But

~


(an' b) ;;:;; b, because for all n we have an b ~ b. Hence

n-l
00

00

b . ~ (an' b) = ~ (an' b) .
n-l

It follows that

n-l

00

00

b • ~ an = ~ (an' b).
n=l

A.3.7. Theorem.
and let

n=l

Let R be a lattice, anE R, (n

=


"
Then the following are equivalent:
00

1.

~ an
n-l
00

II.

E

exists in R.

bk exists in R.
00

In the case of existence of

00

an or of

~
n=l

00


~

n=I

b" we have:

00

~ an = ~ bn .

n=l

n=l

Proof. Let I and put
00

b =df 2: a".

(0)

n-l

By definition of the lattice-sum, we have
an;;:;; b

Hence, taking any k

at


;:s

for all n.

1, we have

< b,

a2;;:;; b,

which gives, [A.2.6.J and by induction,
k

k

n-l

n-l-

2: an ~ 2: b =

(1 )

bk

;;:;;

b,


(e., by (0),

b for all k.

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1, 2, ... ),