Combinatorics Graphs and Algebra [Reprint 2019 ed.] 9783111552255, 9789027975119

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combinatorics graphs and algebra

methods and models in the social sciences

5

MOUTON • THE HAGUE • PARIS

combinatorics graphs and algebra

by CENTRE DE MATHEMATIQUE SOCIALE ÉCOLE DES HAUTES ÉTUDES EN SCIENCES SOCIALES

MOUTON • THE HAGUE • PARIS

This volume has been published with the support of the Department of Social Sciences of Unesco, Paris

ISBN: 90 279 7511 6 © Original French edition: UNESCO Entitled: Combinatoire, graphes et algèbre Published by Mouton and Gauthier-Villars, Paris - The Hague, in collaboration with the École des Hautes Etudes en Sciences Sociales. © English edition: UNESCO, 1976 Translated from the French by: Victoria von Hagen Cover design by Jurriaan Schrofer Printed in the Netherlands

preface

The purpose of this manual, is to introduce students and researchers in the human and social sciences to some mathematical domains of particular importance because of their potential application in these sciences. This first volume is devoted to fundamental chapters in algebra and combinatorics. It has been conceived as an introduction to the meaning and proper usage of certain key words which are encountered more and more frequently in the construction of models or in the presentation of mathematized theories in psychology, sociology, anthropology, linguistics or musical composition. The index of terms found at the end of this manual is thus of major importance: for each term, this index refers the reader to a definition and to the chapters in this text where the term has been used in various contexts. These c o n t e x t s . . . what are they? Those found in the most elementary 'structures' which can be assigned a finite set or a set which can be finitely constructed, structures of ordering, classifications, trees, Boolean algebras, monoids, groups, simplexes and measure scales. Moreover, these topics are now part of what is commonly taught in many universities, especially in French higher education, to students in the social sciences. The various articles in this manual have been written so that they can be read independently: that is, the order in which they can be read is of little importance and is left to the discretion of the reader. Nevertheless, we have made every effort to show the interrelationships that unite the domains introduced in this volume by referring the reader to various other articles. In addition, the bibliography of each chapter furnishes references which will permit the reader to improve on this brief introduction to these mathematical domains, as well as acquainting him with the detailed applications of these topics in the human sciences. The reader is not supposed to have previous mathematical knowledge, other

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Preface

than that normally acquired in high school. However, those readers who encounter too much difficulty can refer to introductory texts such as those already published by certain authors of this manual 1 . In these works the reader will find the elementary bases of combinatorics (subsets of a finite set and their simplicial organization) to which the articles here sometimes refer. The reader can also consult the symbol index placed at the beginning of this manual. M. Barbut

1. M. B a r b u t , C. d ' A d h é m a r , B. L e d e r e and P. Jullien, Mathématiques élémentaires, applications à la statistique et aux sciences sociales, Paris, P.U.F., 1 9 7 3 ; M. Barbut, Mathématiques des sciences de l'homme, Paris, P.U.F., 1968; G. Th. G u i l b a u d , Mathématiques, Paris, P.U.F., 1 9 6 6 ; P. Rosenstiehl and J. Mothes, Mathématiques de l'action, Paris, D u n o d , 1965.

contents

Preface, by M. Barbut

5

Symbol index

9

1. Trees by P. Rosenstiehl and B. Leclerc Introduction 1. Tree words 2. Rooted semi-lattices 3. Modules on trees Bibliography

15 16 31 39 55

2. Ordering and classification by B. Monjardet Introduction 1. Quasiorderings 2. Equivalence relations 3. Orderings 4. Quasiorderings revisited 5. Some applications Bibliography

59 60 68 75 84 87 90

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Contents

3. Boolean algebras, Boolean rings by G. Th. Guilbaud 1. The present status of Boolean algebras 2. Description of a Boolean algebra 3. Boolean rings

91 96 101

4. Simplicial objects by G. Th. Guilbaud 1. The first model: simple complex or simplex 2. Some other models 3. Simplicial category (abstract complex) 4. Simplicial sets (set complexes) 5. Products of simplexes Bibliographic notice

107 112 118 119 121 125

5. Monoids and groups by C. d'Adhemar Introduction 1. Monoids 2. Groups 3. Homomorphisms 4. Permutations Bibliography

127 128 134 139 141 145

6. Measure scales by M. Barbut Introduction 1. Elements of the typology of ordinal scales (types of order) 2. The additive calculus (abelian groups) 3. Additive numerical scales (totally ordered abelians) 4. The additive and multiplicative scales 5. Interval scales. Afflne scales

147 149 152 159 165 171

Bibliography

176

Subject index

178

symbol index

Symbols

Use and meaning

{ }

E = {2,

Commentary (remarks)

b, ft}: the set E and

I E I, card. IE 1 = 4: the cardinal of E is E

Number o f elements in E

four 2 G

2 is an element o f or belongs to

(...,...)

the set E and the list of its elements: 2, n, b, ft

its element

the set E

(x, >»): the ordered p a i r * ) '

starting with a set E we can construct a new unique product set which is the set of pairs (x, y) such that x G E and y G E. The notion of ordered pair has a privileged meaning in set theory, but the number of components can be increased: triplet, quadruplet, . . . n-uplet

EXE = E2

E cross E, or E to the second

the cartesian product of the set

power

E by itself is the set of pairs (JC, y) where x G E and y G E

C

X C Y: the set X is included

X C Y if, and only if, for every x

in the $et Y

we h a v e x e i = * x e Y

10

Symbol index

Symbols

Use and meaning

{x }

the set reduced to a single element x

Commentary (remarks)

X = Y: the set X is equal to the set Y

the equality of the two sets is not satisfied unless they contain the same elements

(V* G X) (x = x): for every x G X we have x = x

the proper use of these two symbols, called logical quantifiers requires a formalism that exceeds the scope of this work; they are only used here as abbreviations in the context indicated here

(3;c G X) (x = y): there exists somex&X such that we have * =y

x => y: x implies y x

if x, then y

x if, and only, if, y

u

J U K I union Y

X U Y designates the set formed by both the elements of X and the elements of y ( a n d of J f a n d Y simultaneously)

P°, CP, P, E-P, QgP

the subset of the set E that is the compliment of the subset P of E

PU P° = E; the union of P and its complementary subset reconstitutes the set E

n

X n Y: X intersection Y

i n y designates the set formed by the elements that belong to both X and Y

W )

the set of the subsets of the set E

if \E I = 4, | VKE) I = 2 4 . Two special subsets are E itself (the whole set) and the empty subset, or t

Symbol index

11

Symbols

Use and meaning

Commentary (remarks)

G(piE). the empty set is an element of the set of the subsets of E

the empty set is the set which contains no elements. It is by definition an element of the set of the subsets of the set under consideration, E

x R y or (x, y)£R

xRy: the pair (x, y) belongs to or satisfies the (binary) relation R

x fly

(x, jv) does not satisfy the relation R

x i=y

x is different from.y

x = y x ^ y x=y (mod R)

x is equivalent to.y x is equivalent to.y, or x is equivalent to.y modulo/?

u

x is equivalent to y from the point of view of the relation R

U ^ , - : union of the family ' 6 / of the sets Xt

the union applies to a family of sets indexed by /

E/R = nR

the quotient set E modulo R the partition induced by the equivalence relation R

the set of equivalence classes of the equivalence relation R is the same thing as the partition of E associated with the relation R

/: E -> F

the mapping / of the set E into the set F

x-*y or x -*• y

y is the value (image) of x by the mapping (function) under consideration

fix)-y

/ o f x equals y

the image of x by the mapping/ is the element y

12

Symbol index Commentary (remarks)

Symbols

Use and meaning

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