The Theory of Partial Algebraic Operations

244 Pages·1997·7.668 MB·English

by  E. S. Ljapin, A. E. Evseev (auth.)

The Theory of Partial Algebraic Operations
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 414
The Theory of
Partial Algebraic
Operations
by
E.S. Ljapin
and
A.E. Evseev
Russian State Pedagogical University,
St Petersburg, Russia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4867-7 ISBN 978-94-017-3483-7 (eBook)
DOI 10.1007/978-94-017-3483-7
This is a completely revised, enlarged and updated
translation of the original Russian Work
Partial Algebraic Operations by the same authors,
Russian State Pedagogical University, 1991.
Translated by 1.M. Cole.
Printed on acid-free paper
All Rights Reserved
©1 997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Contents
PREFACE VII
TRANSLATOR'S PREFACE IX
o
BASIC TERMINOLOGY 1
1 INITIAL CONCEPTS AND PROPERTIES 5
1·1 The Concept of Partial Operation ..... 5
1·2 Partial Groupoids and Their Isomorphisms 8
1·3 Multiplication of Subsets. 10
1·4 Generation ........ . 13
1·5 Weak Associativity . . . . . 16
1·6 Extension of An Operation 23
2 HOMOMORPHISMS 29
2·1 Mappings Preserving Operations 29
2·2 Weak Homomorphisms ..... . 33
2·3 Special Types of Homomorphisms. 36
2·4 Factor Partial Groupoids .... . 43
2·5 Replicas of Partial Groupoids .. . 50
2·6 Pargoid Replicas in Varieties of Total Groupoids 57
2·7 Extensions of Homomorphisms in Total Groupoids 64
3 DIVISIBILITY RELATIONS 69
3·1 Elementary Properties of Divisibility . . . . . . . 69
3·2 Associative Elements with Maximum Divisibility 71
3·3 Ideals ..... . 75
3·4 Active Ideals .. 79
3·5 Long Divisibility 81
3·6 Ideal Chains. . . 84
4 INTERMEDIATE ASSOCIATIVITY 89
4·1 The Concept of Intermediate Associativity. . . . . . 89
4·2 Dividing of Conditions of Intermediate Associativity 91
4·3 Partial Groupoids of Words with Synonyms . . . . . 96
v
vi CONTENTS
4·4 The Closure of the Classes of Semigroups and Groups with Respect
to the Operations of Restriction and Strong Homomorphism 99
5 SEMIGROUP EXTENSIONS OF PARTIAL OPERATIONS 103
5·1 Extensions of Partial Groupoids to Total Groupoids from Some
Varieties. . . . . . . . . . . . . . . . . . . . . . . 103
5·2 Independent Semigroup Extensions. . . . . . . . 110
5·3 Ideal Layers of Semigroups as Partial Groupoids 125
5·4 Commutative Contraction. . . . . . . . . . . . . 131
5·5 The Algorithmic Problem of the Extendability of Partial Operations 137
6 PARTIAL GROUPOIDS OF TRANSFORMATIONS 143
6·1 Transformations and Their Fixed Points . . . . . . . . . . . 143
6·2 The Abstract Characteristic of a Class of Partial Groupoids
of Transformations Complete with Respect to Constant
Transformations ................... 148
6·3 Partial Groupoids of Closure Transformations . . . 155
6·4 Similarity of Partial Groupoids of Transformations 160
6·5 Inner Semigroup Extensions . . . . . . . . . . 163
6·6 Partial Groupoids of Partial Transformations 168
7 FACTORISATION OF PARTIAL GROUPOIDS 175
7·1 Disjoint Factorisation ... 175
7·2 Inflation........... 177
7·3 Annihilating Factorisation . 179
7·4 Cartesian Factorisation. . . 186
7·5 Sub-Cartesian Factorisation 190
7·6 Amalgams of Partial Groupoids . 196
7·7 Semigroup Amalgams ... . . . 200
7·8 Inner Extension of Certain Semigroup Amalgams 207
BIBLIOGRAPHY 215
INDEX 233
Preface
Nowadays algebra is understood basically as the general theory of algebraic oper
ations and relations. It is characterised by a considerable intrinsic naturalness
of its initial notions and problems, the unity of its methods, and a breadth that
far exceeds that of its basic concepts. It is more often that its power begins to
be displayed when one moves outside its own limits. This characteristic ability is
seen when one investigates not only complete operations, but partial operations.
To a considerable extent these are related to algebraic operators and algebraic
operations. The tendency to ever greater generality is amongst the reasons that
playa role in explaining this development. But other important reasons play an
even greater role. Within this same theory of total operations (that is, operations
defined everywhere), there persistently arises in its different sections a necessity
of examining the emergent feature of various partial operations. It is particularly
important that this has been found in those parts of algebra it brings together and
other areas of mathematics it interacts with as well as where algebra finds applica
tion at the very limits of mathematics. In this connection we mention the theory of
the composition of mappings, category theory, the theory of formal languages and
the related theory of mathematical linguistics, coding theory, information theory,
and algebraic automata theory. In all these areas (as well as in others) from time
to time there arises the need to consider one or another partial operation. Such
cases become so numerous that one begins to perceive the need to move from
the individual consideration of the properties of separate concrete cases of partial
operations to that of investigating the framework of a general theory.
The theory of partial operations naturally yields a proper extension of the
theory of total operations. In the present day this area proves to be extremely
ramified and rich, and it is in its period of blossoming. The thought naturally arises
of transferring these concepts and results into a new area of algebra. This, it is
understood, has been a fruitful course followed in many cases. But in the early
stages of constructing a theory of partial operations one finds a need to know
some considerable amount about the specific nature of this path. Often the direct
transfer of results from the theory of total operations proves to be difficult or even
impossible. It proves necessary to make an essential revision or reinterpretation of
the usual algebraic material. In addition, absolutely new concepts and problems
arise which are specific to the new direction pursued. A methodical procedure for
such investigations is therefore required.
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viii PREFACE
And so the theory of partial algebraic operations, being the extension of the
theory of total operations, and making use of the achievements of the latter and the
attempt to apply it at the very frontiers of algebra, has to be formally recognised
as a new and independent path in the vast field of contemporary algebra.
At the moment over a hundred works specifically dedicated to the study of
partial operations have been published. It is impossible to estimate how many
other works on different partial operations have been encountered in the course of
research. In some general algebraic works partial operations are referred to, but
only very briefly.
Up to now there has not been a sufficiently complete and connected presen
tation of the theory of partial algebraic operations. A lack of coordination has
reigned amongst the basic concepts, and even in notation and terminology. It has
been recognised that a presentation has been lacking for the questions that are
needed for the construction of a general theory. Such a state of affairs has greatly
impeded the further development of the theory.
This book has as its goal a coherent presentation of the basic theory of sys
tems with one partial algebraic operation (partial groupoids). The fundamental
concepts and ideas are singled out, and their elementary properties are examined.
After this deeper questions are studied. The fundamental concepts and properties
that are applicable to partial algebraic operations of a general form are presented,
but of course, as in the theory of total operations, subsequent investigations re
quire that they are restricted to various types of partial operation. It is these
considerations that in many places determine the form of this book. In particular,
this applies to the notion of associativity, which in the theory of partial operations
has various forms of expression. The system worked out must offer the possibil
ity of the subsequent continued systematic development of the theory of partial
algebraic operations.
For ease of reading the text the book has been divided into small articles
enumerated by a pair of numbers, the first of which is the number of the section,
and the second the position within that section. This allows cross references to be
made relieving the reader of the necessity of remembering all of the previous text
all the time. For example, the reference 2·3.15 tells the reader that it is in the
second chapter, the third section, and the fifteenth article of that section. Within
anyone chapter the number of the chapter is not cited.
At the end we have provided a Bibliography with a suitably full list of works
devoted to the general theory of partial operations and that are in print at the
time of this book's publication. It should be understood that we have not included
works that refer only in passing to various kinds of partial operations. But we have
included those books and papers which, although they are not related to the theory
of partial operations, are necessary for the reading of this book.
The works are listed, as usual, by author's name and year of publication.
The present book is essentially a broadening of the earlier book (Ljapin and
Evseev, 1991) published in Russian.
E.S. Ljapin A.E. Evseev
Translator's Preface
The author's have graciously liberally assisted with clarifications of the intended
meaning of some difficult portions of text and of their preferred terminology. I
wish here to express my considerable gratitude to them for this help.
The original index has been expanded by the authors, by the creation of a
three level version of it, by the addition of extra citations for many entries, and
by the addition of some extra entries. It is hoped that the resulting index will be
a particularly helpful complement to the text.
Those interested in algebraic topology and differential geometry should know
that the authors' terms layer, envelope, and band are translated from the Russian
words for fibre, covering, and sheaf, with which the constructs here have some
affinity. For those interested in categoriallogic and its applications the construct
here of Cartesian closure has some affinity with the notion of the same name in
the definition of a topos.
It has repeatedly been remarked over the last century that the advances of
algebraic concepts frequently have a later close bearing upon theoretical formul
ations of physical phenomena. The most spectacular have been the applications of
group theory to the classification of crystal structures, group representation theory
to the classification of particles and angular momentum in quantum theory, and
Lie group theory to particle physics. Perhaps the most astonishing has been the
discovery that category theory (known frequently as 'generalised nonsense' in Engl
ish usage and as 'abstract rubbish' in Russian-indicating the general view of the
uselessness of the topic) has been found to be a ready made theory that makes
theoretical computer science immediately tractable and readily comprehended be
cause category theory's extemely compact way of handling complexly nested and
mingled structures of a variety of types gives a ready parallel to the structure of
programming both in software and hardware (including silicon). Neither should
one forget that Clifford created his algebraic structure in an attempt to create a
theory of molecular structures.
In general, physical theories have relied either upon continuous constructs or
upon complete constructs (here the latter have some similarity with the notion of
total operations, in which all relevant constituents are present), but physical reality
is unavoidably incomplete, simply because it is impossible to know everything,
and information is often missing. As much can be said of the attempts over
recent decades to devise adaptive processes that can mimic either apprehending
ix
x TRANSLATOR'S PREFACE
intelligence or learning coupled with application of what has been learnt, or both.
This book is thus of considerable interest for the development of theories that
are based upon information that is incomplete when compared with a 'complete'
model. The results collated in this book are therefore of potential considerable im
portance for identifying the differences between 'complete' and 'incomplete' mod
els and theories, and so allow exploratory inferences of refinements of structure in
those models and theories to be more fruitfully and precisely examined.
The general move to a suspicion that the very foundational problem areas
of theoretical physics may benefit from analyses closer to the nature of category
theory may be considerably helped by this book, therefore. It should, of course,
not be forgotten that the very considerable developments in the application of cate
goriallogic to theoretical computer science and category theory to the categorial
formulation of automata theory over the last two decades lead one to expect that
the study of computer program methods that generate new program constructs in
repeated piecemeal fashion may therefore also benefit from an understanding of
this book's topic of partial algebraic operations.
Work at the forefront of one field that suggests novel ideas in other fields is
always important, and this book has that potential.
Michael Cole