When I was an undergraduate, I used to name my pot plants after my heroes. Spivak, a large fern with delicate, pale green fronds, had pride of place in the living room. Around the rest of the house resided Hartley and Hawkes (one plant or two, I don't remember), Halmos, Jech and other authors of favourite textbooks. Pot plants and toddlers don't mix, so at my new house I intend to plant my favourite fruit tree, an apricot, and name it Louis Rowen.
Louis Rowen's new book is designed as a text for a year-long introductory course in abstract algebra. As its title suggests, this book covers three main areas - groups, rings and fields. I would recommend it highly for strong second-year students - the type of class you would give Spivak rather than Thomas & Finney as a Calculus text. Rowen acknowledges the influence of Herstein's Topics in Algebra , and while having many similarities in content and structure, does not cover any linear algebra. I regard this as a strength, for I could not imagine ever reading Herstein cover-to-cover. I believe no maths book should be more than about 200 pages long.
Part I deals with groups. Starting with the definition of a monoid, it progresses to Sylow's Theorems and soluble groups. Throughout this section, he uses the symmetric group S_n as his main example - to both motivate and illustrate the theory. This works really well. As the theory is developed, he classifies all groups of order less than 60. Further topics, such as group actions, nilpotent groups, classical groups and wreath products are developed in the Exercises. Part I alone would make an excellent one-semester group theory course.
Part II is on rings. The climax of this section is a proof of Fermat's Last Theorem for n=3. Formal power series, UFDs and Nagata's Theorem are developed in the exercises. Again, the plot is driven by one main character - the polynomial ring F[x] over a field F. All the way through the book, Rowen keeps reminding us of the origins of abstract algebra in number theory. I particularly like the way he introduces Noetherian rings by talking about the factorization of the number 48 into primes. Although Rowen succeeds quite well in leading the student through what can be a confusing jungle of PIDs, UFDs, integral domains, etc., his initial definitions should have been less confusing. The poor student who forgets two chapters later what an integral domain was in the first place, has to trace its definition all the way back to that of a monoid. Also, the definitions of groups and integral domains both come at the end of a paragraph, whereas they should be out in the open, surrounded by flashing neon lights.
Part III on fields covers constructions with straight edge and compass, finite fields and solvability of equations by radicals. I feel that this section suffers from the lack of a well-defined ``main character''. Examples of field extensions and their Galois groups are given, and the cyclotomic extensions are treated in detail, but these are not referred to when normal extensions are defined, or when the Galois correspondence is introduced. This part of the book seems rather rushed. Of course, there is great incentive to get to the beautiful applications of the theory to finite fields and the insolubility of the quintic by the end of the course. However, I would prefer to include discussion of how the theory relates to a ``main character'', even if it means covering the theory in a little less depth (perhaps concentrating on characteristic zero).
The book begins with a quote from Leibniz. \item{} {...This great science which I have been calling `` Characteristic,''\ of which Algebra and Analysis are but small branches, ... is what gives words to languages, letters to words, digits to Arithmetic, notes to Music; this is what teaches us the secret of precise reasoning, requiring us to leave, as visible traces on paper, a volume for inspection at leisure: finally, this is what makes us reason, substituting characters in place of things, thereby unburdening the imagination.}
Why don't mathematicians any longer write like that about mathematics? On page 141, Rowen writes \item{} {Possibly by the time this book has appeared, the proof set forth by Wiles will have been found to be complete. This might cause a let-down, because one of the great quests of mathematicians would be complete. However, one should view as a great tribute to the human spirit, that the final solution of Fermat's Last Theorem would come before World War III or any other final solutions.}
This is nowhere as poetic as Leibniz, but I find it just as astonishing - not for the political reference, but because an undergraduate textbook is referring to how mathematicians feel about a theorem being proved. I have only ever encountered one maths-book-with-attitude, namely Miles Reid's Undergraduate Algebraic Geometry . Attitude is a spice for writing - it shouldn't be overpowering, but without it everything is dull. If I were dictator, every author and every lecturer would have to tell their class which is their favourite theorem in the book/course and why, and also state what part they like least, and why it is included anyway.
Rowen's book, particularly the discussions of finite simple groups, Noetherian rings and number theory, tells the student that algebra is a dynamic, alive and growing subject, not something that was all worked out by Cauchy and Cayley, then finished by Noether. He mentions in various places connections with areas such as character theory and algebraic geometry, indicating that there are further delights beyond this first course. He includes an appendix on noncommutative rings (an area with which I am unfamiliar), as a `` luxury''. Rather, I would regard it as essential that mathematicians talk to undergraduates about our own areas of special mathematical interest.
One of the great delights of mathematics is the way beautiful theorems so often have so many different proofs. By seeing several proofs, we not only understand the theorem better, but we also see the unity of mathematics. Rowen says he tries to chose the most memorable (I would call it the most illuminating) proof; and he puts the other proofs in the Exercises. He also warns that the exercises are mostly extensions of the text, rather than routine applications of the main results. Indeed, these Exercises could be used in a reading courses to third or fourth year students.
When the gremlins attacked this book, they didn't do very much, but they picked their targets well! In the definition of a ring, they removed the word `` abelian''\ from ``(R,+,0) is an abelian group''. A few pages later, in Note 4, they removed the word `` no'' from `` \ldots R is a field iff R has no nontrivial ideals''. There is the occasional missing finiteness hypothesis, such as on p.83, where he states that the classification of simple groups has been finished, and in Theorem 9.2 and Exercise 10.1. Most of the errors I detected were in the Exercises, and most of these mistakes were quite minor, (Exercises 2.12, 6.10, 8.4, 10.1, 14.8, 16.14, 16.15, 21.13, 22.11, 23.5, 23.6, 23.14, 24.3 and 27.17.)One very annoying feature of my copy of the book is that from the start it naturally opened at page 58, which is a very ugly example of bad mathematical typsetting. Generally, the typesetting leaves much to be desired.
I can find very few faults with the exposition. In Chapter 7, I was a little confused by some of his proofs until I realized he was allowing redundant generators and/or trivial summands. This might be made explicit. Also, there is minor confusion in the proof of the first part of Sylow's Theorem where he refers to a subgroup of order p^t as a Sylow subgroup, where it is only Sylow if t is maximal. I would also have chosen to mention explicitly that \{0\} is an ideal.
In summary, what I like best about this book is that it has a hint of attitude. I would like there to be more, but I am not used to seeing any. And what I like least is that definitions and other important statements are sometimes buried in the text in a way that makes them difficult to refer to.
Royal Melbourne Institute of Technology
In the opening sentence of the preface to Introduction to Lattices and Order B.~A.~ Davey and H.~A.~ Priestley claim, ``This is the first textbook devoted to ordered sets and lattices and to their contemporary applications.'' For this review I examine the force of this claim and, on the basis of my classroom experiences, evaluate the usefulness of the book
The phrase ``ordered set'' appearing in the claim refers to what in previous years was called a ``partially ordered set.'' Recall that a lattice is a special type of ordered set - one in which every pair of elements has a least upper bound and a greatest lower bound. The algebraic definition is that a lattice is a set with two binary operations that are each commutative, associative, idempotent, and satisfy an absorption law. Contemporary applications of lattices and ordered sets of course vary over time. Applications that were contemporary some time ago need not be of interest today. Currently some of the most exciting applications of lattices and ordered sets are in certain subareas of computer science, social science, and operations research.
A keyword appearing in the claim is ``textbook.'' Many an author who has written an advanced book on mathematics has subsequently viewed the work as an appropriate textbook for a course, and may then have actually used the book in such a capacity when given the opportunity. But for the most part such books are not textbooks. They are not written with the student and classroom use foremost in mind and their level and pace are not suitable for undergraduate or beginning graduate students.
When one thinks of books on lattice theory, the first to come to mind is Birkhoff's Lattice Theory , which was also the first to be published. The first edition appeared in 1940 (155 pages), the second in 1948 (283 pages), and the much enlarged third edition appeared in 1967 (418 pages), and it is still in print as volume 25 of the American Mathematical Society Colloquium Publications. Birkhoff's book is comprehensive in its coverage of lattice theory and, even almost thirty years after publication of the third edition, it remains a most valuable resource. The theory of ordered sets is certainly not a primary concern of Birkhoff, but buried in the asides and footnotes of his book are many important results involving ordered sets other than lattices. Much of Birkhoff's book is devoted to applications of lattice theory to other areas of mathematics. Thus there are chapter titles such as ``Applications to General Topology'', ``Applications to Logic and Probability'', and ``Borel Algebras and Von Neumann Lattices''. However, it would be inappropriate to call Birkhoff's encyclopedic Lattice Theory a textbook.
Birkhoff's Lattice Theory did prompt others to write textbooks on the subject. Fine books such as Introduction to Lattice Theory by G.~ Sz\'asz, Lattice Theory by T.~ Donnellan, and Introduction to Lattice Theory by D.~E.~ Rutherford appeared in the 1960s. These are suitable as undergraduate textbooks and provide a solid introduction to the algebraic theory of lattices. But they fail to satisfy the Davey and Priestley claim in that either they contain no applications or devote only a few pages to ordered sets.
In the 1970s several advanced graduate texts and research monographs on lattice theory appeared. These books included not only many of the by now classical results found in Birkhoff's three editions, but also chronicled the many new results obtained after 1965. For example, R.~Balbes and Ph.~Dwinger's Distributive Lattices is a thorough study of distributive lattices and the applications of the theory of these lattices to algebras associated with non-classical logics. The text Algebraic Theory of Lattices by P.~Crawley and R.~P.~Dilworth contains many very deep and difficult results at the ``heart'' of the theory of the structure of lattices. G.~Gr\"atzer's two books Lattice Theory: First Concepts and Distributive Lattices (1971) and General Lattice Theory (1978) provide a comprehensive study of general lattice theory with special emphasis on equational theory and free structures. All the books mentioned in this paragraph focus on lattices and give little emphasis to ordered sets. The applications presented in them are primarily to algebra or to algebraic logic.
In the past twenty years research on ordered sets other than lattices and applications of ordered sets to ``contemporary'' topics such as scheduling theory, combinatorics, and algorithms has burgeoned. The journal Order , for which Davey and Priestley are members of the editorial board, chronicles this burgeoning research. Ordered sets are now an obligatory topic in discrete mathematics textbooks, and appear, for example, as a chapter topic in R.~Stanley's Enumerative Combinatorics . The books Linear Orderings by J.~Rosenstein and Combinatorics and Partially Ordered Sets by W.~Trotter deal in depth with certain special topics in the area and do not deal with the theory of lattices. Thus, prior to the publication of Introduction to Lattices and Order no textbook ``devoted to ordered sets and lattices and their contemporary applications'' had appeared.
How well does the Davey and Priestley book live up to the opening sentence of its preface? Introduction to Lattices and Order is certainly a textbook. It is carefully organized into sections and subsections. It is loaded with examples, many of which reappear as needed throughout the book. Each chapter has a battery of exercises of varying difficulty. Sprinkled throughout are paragraphs labelled ``Stocktaking'' in which the authors attempt to summarize what has been done so far in the book and to place the material in a broader context. The level is suitable for advanced undergraduates. Topics that that might be unfamiliar to undergraduate readers are developed as needed. Thus, a section ``Topological Toolkit'' presents the topology required for the chapter on duality theory.
The authors give elementary lattice theory a thorough treatment. Their emphasis is on distributive lattices and complete lattices. This selection is in accord with the authors' research interests and their choice of applications. The representation theorems for distributive lattices are presented in detail. Also included are the by now fairly standard applications of Boolean algebra to switching functions and propositional logic.
Ordered sets appear in several settings in the Davey and Priestley book. There is a chapter devoted to Complete Partial Orderings (CPOs) and their applications to information structures and to denotational semantics. Another chapter deals with fixed points in ordered sets. The authors hint at the applications of fixed point theory to program verification. A very slick discussion of Zorn's Lemma and its equivalents appears at the end of this chapter. Another application, Concept Analysis, is the subject of the final chapter. This is a recently developed order-based methodology for organizing hierarchies of concepts.
I used Introduction to Lattices and Order as the sole textbook in a one semester course. The students enrolled were a heterogeneous group including modestly prepared undergraduates, well trained graduate students, and a few applications-oriented computer science students. The undergraduates found the book challenging yet manageable; the graduates would have benefited from some more challenging exercises but they enjoyed the elegant proofs; and the computer science students actually appreciated the CPO and fixed point material and commented on how the presentation in the book clarified things for them. In short, the textbook was a success.
My classroom use of Introduction to Lattices and Order was with the original 1990 edition. This printing contained the number of typographical and other errors typical of any first edition. Cambridge University Press has reprinted this edition three times. The reprintings have corrected all such errors that I am aware of, and have also subtly modified other parts of the book without changing the pagination. A fine example of these micro revisions is the discussion of the cardinality of the free distributive lattice generated by a set X of free generators. In 1990 the largest value known for the size of these lattices was a 13 digit number corresponding to |X|=7 and the 1990 printing states that the cardinality for ``|X|=8 remains elusive.'' In 1992 the 23 digit number giving the |X|=8 value appeared in an article in Order . In the 1994 printing this number is presented and the book now reads ``|X|=9 remains elusive.'' Thus, the Davey and Priestley book is contemporary and remains a first choice for a textbook on lattices and ordered sets.
University of Illinois at Chicago
USA
In his introduction Dieudonn\'e states that the book is written for those who are not professional mathematicians, to give some appreciation of the thrust of current mathematical activity. He assumes that the reader has knowledge up to the level of the French scientific baccalaureate and he consciously keeps his many examples to the constraints of this background adding appendices for amplification. However, I feel that the readers who would best appreciate this book are those who have familiarity with some of the areas he discusses. The book enlarges perspective on the wide field of mathematical endeavour and enhances understanding of the essential unity of mathematics. I would recommend this book as a general text and required reading for all graduating mathematics students.
The work is more than a history of mathematics, but it does cover the main mathematical advances of the last two centuries. As might be expected from such a master as Dieudonn\'e, his is the story of the great movements in thought which have moulded mathematical activity to give it the shape it has at the present time.
The program he sets himself is to reveal how the abstract nature of mathematical objects in our day flows from the methods invented between 1800 and 1930 to solve classical problems. He aims to show that the usefulness of those objects is greater today than ever it was.
The dominant creative theme is the hypothetico - deductive method described by Plato and put into practice by Euclid. This was given a totally precise form around 1900 by Pasch and Hilbert for geometry and was almost at once extended to all branches of mathematics to become the routine of today's mathematics.
The first two chapters are rather light and are by way of introduction. The first chapter gives background on mathematicians and the mathematical community. It contains a rather challenging definition of a mathematician as ``someone who has published at least one non-trivial theorem''. The second chapter is an acknowledgement of the place of applied mathematics with its obvious successes in the eighteenth century supplying mathematical models to give rational explanations of natural phenomena which are the subject of other sciences. However, it comes as no surprise when he asserts that this is not the chief thrust of contemporary mathematics.
The main thesis of the text is contained in the three chapters which follow. He begins by discussing the essential features of Greek mathematics whose fundamental characteristics are the idea of proof and the awareness that the objects of mathematical study are immaterial entities gained by abstraction from objects accessible to our senses. He points out that the difficulty in relating abstract objects to the reality which they picture led to the statement of axioms to define the mathematical objects.
Alongside this system which originated with geometry there developed the logistic tradition for numerical calculation which took shape under Diophantus. The difficulties of the geometers with incommensurable magnitudes and the algebraists with equations having no rational solutions, were to be solved by the creation of new mathematical objects.
The importance of approximation procedures, ideas which originated with Greek mathematics was generalised in succeeding centuries. Cauchy's `Axiom of nested intervals' asserted the existence of the real number being approximated and became the linchpin on which analysis turns. The introduction of the `Method of co-ordinates' by Descartes linking algebra to geometry was the first of several bridges linking dissimilar areas of mathematics and a first successful demonstration of the fundamental unity of mathematical ideas.
The nineteenth century was one of astonishing productivity. The renaissance began with Gauss and the French school which developed at the Revolution. The century saw a change in style and content with a `return to rigour' and the introduction of new mathematical objects and methods. Besides the discovery of new concepts, greater depth was given to old ideas. Early in the twentieth century the idea of structure was seen to be important and the unity of the subject was affirmed when it was realised that very different theories shared the same underlying structure. He gives many examples to illustrate this: the discovery of complex numbers and vectors, the examination of algebraic structures such as groups and of topological structures such as metric spaces, the development of set theory and the comparison of structures by isomorphisms.
In chapter six he states that the important feature of the nineteenth century development was the progressive abandonment of the concept of `self evident truths'. He illustrates this principally by the development of non-euclidean geometries, the axiomatisation of the number system and the analysis of infinite sets. He discusses the Zermelo-Fraenkel system and the recognition of the role of the Axiom of choice and G\"odel's theorem on the incompleteness of arithmetic.
Besides a general index there is an historical index giving information about the mathematicians cited in the book. Glancing at his alphabetical list, the reader might be shocked at some omissions; I found no reference to Baire and for me his work is an important tool. Very little is said about Lebesgue's work although he is mentioned. But then you have to remember the constraints that Dieudonn\'e set himself: to communicate with the reader who has less than a professional concern with mathematics.
Jean Dieudonn\'e is well known through his many volumes entitled Treatise on Analysis . But he was also a founding member of the Bourbaki team and contributed volumes to its series. I once met him at a party given in his honour when he was visiting a Canadian University. The new English edition of Bourbaki's {\sl Topological Vector Spaces} had just appeared and I remember his embarrassment when he was asked to sign a copy.
In the first chapters of the book he discusses Hardy's opinion that ``mathematics is a young man's game''. But it is interesting to note that the 1987 French edition, Pour l'honneur de l'esprit humain , was published when Dieudonn\'e was in his eighty second year! What a great finale to a brilliant mathematical life: to look back over the great edifice which is modern mathematics to which he had contributed so significantly and to discern for students of the next generation the motivation and great themes of contemporary mathematics.
The University of Newcastle
Presented in this book is a collection of computation problems which author encountered during his tenure as Chief Scientist at NeXT, Inc., and from his scientific computation lectures in the Department of Physics at Reed College.
The format followed in the book is \text{Chapter - Subject - Project}. Each chapter introduces a different area of computational interest which is further subdivided and discussed in the subject subsections. The chapter areas addressed are : Selected Topics in Numerical Analysis, Exploratory Computations in the Sciences, Number Theory, FFT and variants, Wavelets, Chaos and Fractals, and Signal Processing. Following a brief subject area description, each section contains several, multiple part, projects expanding upon the section theme. For example, within the FFT chapter the `subjects' considered are : Discrete Fourier transform (4 related projects), FFT algorithms (7 projects), Real-valued transforms (4 projects), Number-theoretic transforms (1 project). Each project has an associated difficulty scale provided by the author ranging from \text{Level 1 - Introductory}, through to \text{Level 4 - Exploratory research problem, perhaps intractable}.
The book has several noteworthy strengths. It contains a remarkable collection of interesting projects at various indicated levels from topics in science and engineering which are of current interest. The sometimes brief discussions of the `subjects' are accompanied by a reference to the literature for a more detailed description of the material. Presented in the appendix as \text{Mathematica} or \text{C} source code listings, or on the 3.5 DOS-formatted disk accompanying the book, are many of the algorithms described in the projects. These features make it very attractive as a classroom resource.
The algorithms and brief discussions also make this book a useful research tool. This reviewer found the algorithms for computing special functions to be particularly enlightening. In addition, the brief introduction and subsequent projects allows a reader to quickly gain a feel for some of these important areas of scientific computing, and the techniques currently being employed.
The major weakness of the book would be one of omission. With little difficulty we could all berate the author for specific subjects omitted. For example, in the section titled `Equation solving', regarding the solution of systems of linear equations, the author remarks: ``Indeed, multiplication and inversion of matrices are fundamental to large (linear) system computation.'' No discussion is given to the use (or even existence) of modern iterative methods for the solution of large linear systems!
Projects in Scientific Computation is a valuable reference book for its brief discussions and algorithms of several modern topics, and a valuable teaching resource for projects in scientific computational courses.
Clemson University
USA