Semirings and Affine Equations over Them: Theory and Applications

242 Pages·2003·9.32 MB·English

by  Jonathan S. Golan (auth.)

Semirings and Affine Equations over Them:
Theory and Applications
Mathematics and ItsApplications
Managing Editor:
M.HAZEWINKEL
CentreforMathematicsand Computer Science,Amsterdam,TheNetherlands
Volume556
Semirings and Affine
Equations over Them:
Theory and Applications
by
Jonathan S. Galan
DepartmentofMathematics,
UniversityofHalfa.Halfa,Israel
Springer-Science+Business Media, B.V.
AC.!.P.Catalogue record forthisbookisavailable from theLibrary ofCongress.
ISBN978-90-481-6310-6 ISBN978-94-017-0383-3 (eBook)
DOI 10.1007/978-94-017-0383-3
Printedonacid-free paper
AllRights Reserved
© 2003SpringerScience+BusinessMediaDordrecht
OriginallypublishedbyKluwerAcademicPublishersin2003.
Softcoverreprintofthehardcover 1stedition2003
Nopartofthisworkmay bereproduced, stored inaretrieval system,ortransmitted
inany formorbyany means, electronic,mechanical, photocopying,microfilming,recording
orotherwise, without written permission fromthe Publisher,withtheexception
ofany material supplied specifically forthe purpose ofbeing entered
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Contents
ix
Preface
Introduction Xl
Chapter 1: Semirings 1
Chapter2: Partially-Ordered Semirings 27
Chapter3: Comp1eteSemirings 39
Chapter4: Residuated Semirings 49
Chapter5: Matrix Semirings 59
Chapter6: Symmetrie Extensionofa Semiring 81
Chapter7: Semimodu1es 101
Chapter8: Homomorphisms between Semimodules 115
Chapter9: Affine Maps between Semimodu1es 129
Chapter 10: Partially-ordered Semimodules 137
Chapter 11: Eigenelements 151
Chapter 12: Permanents and Determinants 175
Bib1iography 191
Index ofApplications 225
Index ofTerminology 229
Preface
So long as a man remains a gregarious and sociable being, he
cannot cut himself offfrom the gratijication of the instinct of im
parting what he is learning, of propagating through others the ideas
and impressions seething in his own brain, without stunting and at
rophying his moral nature and drying up the surest sources of his
future intellectual replenishment.
- J.J. Sylvester
This volume is a direct continuation of the line of thought presented in my
previous books, Semirings and Their Applications [215], and Power Algebras
over Semirings [216]. It was originally conceived as a portion of a much more
ambitious work on linear algebra over semirings, to have been written jointly
with Prof. Kazimierz Glazek. For various reasons, however, the intended
collaboration could not be carried out at this time, and so we decided that I
should publish this part on my own. I wish to express and acknowledge my
debtto Professor Glazek notonly forsuggestingthe topic, butalso forproviding
a very fine and extensive list ofsources in this rapidly-growing area of semiring
theory and for his comments on some ofthe earlier drafts ofthis material. His
more general bibliographies, [201] and [202] were, needless to say, invaluable
research tools.
Many thanks are due to the UniversityofIowa, at which Iwasa visitingpro
fessorduring the 2002/3academicyear, forprovidingme withfinancial support,
theresourcesand theatmosphereformathematicsresearch. Ioweaspecialdebt
of Prof. Daniel D. Anderson, Prof. Charles Frohman, and Dr. Olga Sokratova,
with whom Ihad manyenrichingandenlighteningconversationsabout material
which, in the end, found its way into this book.
Similarly, I am greatly indebted to Prof. Yefim Katsov of Hanover College,
who visited me in Haifa andwhom I later visited in the course ofmy sabbatical
year, for interesting and stimulating conversations.
Many thanks are also due to the University of Haifa for providing me with
a sabbatical leave that allowed me to devote my time to research and writing.
ix
x PREFACE
Finally,the biggest debt by far is that owed to my wifeand family, who had
to put up, patiently and resignedly, with Yet Another Book.
Introduction
In theory, ihere is no difference between theory and practice; in
practice there is.
- YogiBerra
Semiring theory stands with a foot in each of two mathematical domains.
On one hand,semirings areabstract mathematicalstructures and their study is
partofabstractalgebra- arising abinitiofrom theworkofDedekind,Macaulay,
Krull, andotherson thetheoryofideals ofacommutativering andthen through
the more general work of Vandiver - and the tools used to study them are pri
marily the tools of abstract algebra. On the other, the modern interest in
semirings arises primarily from fieldsofapplied mathematics such as optimiza
tion theory, the theory of discrete-event dynamical systems, automata theory,
and formal language theory, as wellas from the allied areas oftheoretical com
puter science and theoretical physics,and the questions being asked are,for the
most part, motivated by applications.
The aim of the previous books, Semirings and Their Applications [215],
and Power Alqebras over Semirings [216], was to build a bridge between these
two domains, and this volume is intended to continue towards that end. Its
specific purpose is twofold. The first objective is simply to present and em
phasize a portion of the considerable corpus of new work on semirings which
has appeared since [215] and [216], and in particular to bring to the attention
ofthe mathematically-minded reader some of the more interesting applications
to which semiring theory is now being put - by computer scientists, systems
analysts, physicists, and others. The second objective is more technical. In
the concluding chapters of [215] and its predecessor [211], I maintained that
most important applications of semiring theory can be reduced, in the end, to
problems offinding fixed pointsofaffinemaps from asemiring R to itself; that
isto say, given elements a and b ofa semiring R, our problem is to find the
set ofall elements r of R satisfying r = ar +b or r = ra +b. Ifthis set is
nonempty weare interested in knowing ifit is a singleton,i.e, ifOUf affine map
has a uniquefixed point. Ifit does not, we may be interestedin locating certain
designated subsets of the set of its fixed points, for example - in the case R
Xl
xii INTRODUCTION
is partially ordered - the minimal or maximal fixed points may be of particu
lar importance. In this volume, I wish to take another look at semirings and
semimodules over them, with an eye on creating the mathematical framework
to deal with a more general version of this problem: given a semiring R, left
R-semimodules M and M', R-homomorphisms a and ß from M to M',
and elements u and y of M', we would like to see if we can identify the
set of all elements m E M satisfying the condition that ma +u = mß+y.
Again, we can ask the same sort ofquestions:
1. Is this set nonempty?
2. Ifit is nonempty, is it a singleton?
3. Ifit is nonempty but not a singleton, can we identify subsets of it which
are of particular importance?
Of course, we do not expect to be able to answer these questions completely,
except perhaps in special cases. Indeed, if Xl," " X are indeterminates
n
over the semiring of nonnegative integers N, then N[XI, ...,Xn] is also a
semiring and Q[X1,••.,Xn] is a semimodule over it. A general algorithm to
find solutions to the above problems would imply a positive answer to Hilbert's
Tenth Problem, contrary to the well-known result of Matiyasevich. However,
even partial solutions, especially those accompanied by effective computational
techniques, are ofsignificant importance and have many applications.
The book is divided into twelve chapters. The first of these is introduc
tory, introducing manyof the basic notions about semirings already introduced
in [215]. However, these are bolstered by a wealth of new and recent exam
ples and applications, as well as some new constructions which we will need in
the ensuing chapters of the book. In Chapter 2, we study partially-ordered
semirings. Again, this material builds on results from [215], but extends them
to fit our needs here. Of particular interest is the notion of a weak uniquely
difference-ordered (WUDO) semiring, developed, in his thesis, by my student
Wu Fuming [480], which has proven to be an extremely valuable tool in appli
cations. Another very interesting idea presented there is the work of Gaubert
and Katz [187] on the Presburger logicoveradditively idempotent commutative
semirings. Chapter 3 concerns itself with complete semirings, which are be
coming more and more valuable in theoretical computer science, while Chapter
4 presents a short summary of residuated semirings, with an emphasis on the
relation between residuated lattice-ordered semirings and MV-algebras. We
do not pursue the relation between semiring theory and more general cases of
Lindenbaum algebras, but the reader is urged to consider it.
Semirings of square matrices over a given semiring are discussed in
Chapter 5. We take this opportunity to present some of the beautiful re
cent results by Il'in. In Chapter 6, we present and develop the notion of the
R
symmetric extension of a semiring R. This construction was originally
developed by Gaubert and others in the Max Plus Working Group at INRIA