Power Algebras over Semirings: With Applications in Mathematics and Computer Science

207 Pages·1999·7.643 MB·English

by  Jonathan S. Golan (auth.)

Power Algebras over Semirings
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 488
Power Algebras over
Semirings
With Applications in
Mathematics and Computer Science
by
Jonathan S. Golan
Department of Mathematics,
University of Haifa,
Haifa, Israel
Springer-Science+Business Media, B.V.
Library of Congress Cataloging-in-Publication Data
ISBN 978-90-481-5270-4 ISBN 978-94-015-9241-3 (eBook)
DOI 10. 1007/978-94-0 I 5-924 1-3
Printed on acid-free paper
All Rights Reserved
© 1999 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1999.
Softcover reprint of the hardcover 1st edition 1999
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.
to Theresa,
whose friendship was everything
Table of Contents
Preface ix
Some (hopefully) motivating examples 1
Chapter 0: Background material 7
Chapter 1: Powers of a semiring 27
Chapter 2: Relations with values in a semiring 37
Chapter 3: Change of base semirings 61
Chapter 4: Convolutions 67
Chapter 5: Semiring-valued subsemigroups and submonoids 89
Chapter 6: Semiring-valued groups 113
Chapter 7: Semiring-valued submodules and subspaces 141
Chapter 8: Semiring-valued ideals in semirings and rings 155
References 169
Index 189
Preface
This monograph is a continuation of several themes presented in my previous
books [146, 149]. In those volumes, I was concerned primarily with the properties
of semirings. Here, the objects of investigation are sets of the form RA, where
R is a semiring and A is a set having a certain structure. The problem is one of
translating that structure to RA in some "natural" way. As such, it tries to find
a unified way of dealing with diverse topics in mathematics and theoretical com
puter science as formal language theory, the theory of fuzzy algebraic structures,
models of optimal control, and many others. Another special case is the creation of
"idempotent analysis" and similar work in optimization theory. Unlike the case of
the previous work, which rested on a fairly established mathematical foundation,
the approach here is much more tentative and docimastic. This is an introduction
to, not a definitative presentation of, an area of mathematics still very much in
the making.
The basic philosphical problem lurking in the background is one stated suc
cinctly by Hahle and Sostak [185]:
". .. to what extent basic fields of mathematics like algebra and topology
are dependent on the underlying set theory?"
The conflicting definitions proposed by various researchers in search of a resolution
to this conundrum show just how difficult this problem is to see in a proper light.
I try to acknowledge this fact by often indicating the existence routes which I do
not intend to pursue at this moment, but which should serve as an invitation and
challenge to the reader.
Since the development of a new mathematical theory is basically inductive -
one begins with a large number of instances which appear in various mathematical
contexts and tries to develop a general abstract framework in which to understand
them best, I have tried to give a large number of examples, taken from various
areas, to illustrate where the ideas here originated. These are not pursued in
IX
x ____P OWER ALGEBRAS OVER SEMIRINGS ____
detail, but citations to the literature will allow those interested to study them in
greater depth.
Much of the material in this work was originally arranged for presentation at a
seminar on semirings which I directed while a visiting professor at the University
of Idaho during the 1997/8 academic year. In addition to me, the participants in
the seminar included professors Erol Barbut and Willy Brandal, and the graduate
students Lixin Huang, and Minglong Wu. lowe them - and the various other
faculty members and students whom, from time to time, I managed to waylay
in the halls - many thanks for their comments and suggestions and for patiently
following me down several mathematical dark alleys in the search of the tao of
power algebras. The references chosen were intended to illustrate my approach,
and should in no way be considered a comprehensive survey of the literature. In
all probability, more relevant results have been left out than have been included.
Special thanks are due to the University of Haifa for granting me sabbatical
leave during the 1997/8 academic year and to the Department of Mathematics
at the University of Idaho for their warm hospitality and for arranging, and par
tially funding, my very enjoyable stay in that beautiful and tranquil area of the
United States. Similarly, thanks are due to the Departments of Mathematics at
the University of Minnesota in Minneapolis/St. Paul, the University of Tennessee
in Knoxville, and Rutgers University in Piscataway, for arranging, and partially
funding, my stay at their respective institutions during the summer of 1998.
Some (Hopefully) Motivating
Examples
In order to motivate the topics discussed in this volume, we begin with several
examples.
Let A be a non empty set. It is well-known that there is a bijective correspon
dence between the family of all subsets of A and the family lEA of all functions from
A to lE = {O, I}, which assigns to each subset B of A its characteristic junction
I if a E B
{ °
XB:al-t
otherwise.
The set lE of course, has the structure of a complete bounded distributive (in fact
linear) lattice with respect to the operations V (supremum) and 1\ (infimum) and
with the induced partial order::; given by 0 ::; 1. If we think of lEA as a direct
product of copies of lE, then we see that these operations and this partial order
carryover to lEA by componentwise definition:
(1) If U ~ lEA and if a E A then
(VU): a I-t V{x(a) liE U}
and
(I\U): a I-t I\{x(a) liE U}
(2) If I, /' E lEA then I::; /' if and only if I(a) ::; /,(a) for all a E A.
As a consequence of these definitions, it is easy to see that if {Bi liE O} is a
= =
family of subsets of A and if we set C UiEflBi and D niEflBi then
V
=xc
XBi
iEfl
J. S. Golan, Power Algebras over Semirings
© Springer Science+Business Media Dordrecht 1999
2 ____P OWER ALGEBRAS OVER SEMIRINGS ____
and
AXB. =XD·
iEfl
Moreover, if Band B' are subsets of A then B ~ B' if and only if XB :::; XB'.
These observations, which should be familiar to every undergraduate mathemat
ics major, lead to interesting generalizations, which have their origins in several
variations on simple Cantorian set theory. These took several forms:
(I) EXAMPLE. It is sometimes very important to allow an element of a set to
appear in that set "more than once". This has led to the theory of multisets, which
were first introduced by Donald Knuth [209] for use in computer science and have
since been used extensively in many contexts. Thus, given a nonempty set A, a
multisubset of A is defined by a multiplicity function in r::JA, where r::J is the set of
all nonnegative integers. The theory of multisets has been formalized in [42]. For
a formalization of linear logic in terms of multisets, refer to [13] and [382].
(II) EXAMPLE. Loeb [251]' concerned with various combinatorial problems,
extended the notion of a multiset to that of a hybrid set, or "set with a negative
number of elements" by considering multiplicity functions belonging to ;Z;A. Also
refer to [71]. For the use of hybrid sets in the construction of colored Petri nets,
refer to [191]. Another extension of the notion of a multiset involves looking
at multiplicity functions in RA, where R = r::J U {-oo, oo} and where the usual
addition and multiplication of r::J are augmented in the following manner:
-00 + r = r + (-00) = -00 for all r E R;
00 + r = r + 00 = 00 for all - 00 #- r E R;
r . 0 = 0 . r = 0 for all r E R;
-00 . r = r . -00 = -00 for all 0 #- r E R;
00 . r = r . 00 = 00 for all 0, -00 #- r E R.
Elements of RA are sometimes called bags on A. (On the other hand, the term
"bag" is often used as a synonym for "multiset", so one has to be careful.) See
[19] for an application of this construction to signal processing and [411] for an
application to the modeling of fuzzy systems.
(III) EXAMPLE. Zadeh [416], in his ground-breaking work, enlarged the con
cept of a subset further. Given a nonempty set A, a fuzzy subset of A is defined
by an extent of membership function in II A, where II is the unit interval on the
real line. The theory of fuzzy sets, which has since spawned an extremely large
mathematical and engineering literature and which has led to many interesting